Morsi backers also showed their defiance by briefly setting up a third camp near the airport, but later folded their tents and left.
The military-backed interim government seeks to end a political stalemate that has paralyzed Egypt and deeply divided the country. Supporters of Morsi and his Muslim Brotherhood say they will not disperse until he is returned to power.
The second-ranking U.S. diplomat arrived in the Egyptian capital for talks on the political crisis, as Secretary of State John Kerry warned both sides that "the last thing we want is more violence."
Also Friday, Amnesty International reported cases of alleged killings and torture at the hands of Morsi supporters inside the protest camps, saying that one man had his throat cut and another was stabbed to death.
In southwestern Cairo, police fired tear gas at Morsi supporters who rallied in front of Media City, a site housing most of Egypt's private TV stations, a security official said. A second official told the state news agency that protesters tried to "obstruct traffic in an attempt to affect work at the complex."
The rally was "a desperate attempt by rioters from the (Islamist) current," Maj. Gen. Abdel-Fattah Othman, a spokesman for the Interior Ministry, told the private TV station Mehwer. "There was reinforcement from police and army that will not allow any reckless person to get close to the Media City or storm it."
He described the protesters as "brainwashed" to attack broadcasters perceived as secular opponents of the Islamists. Last year, Morsi supporters held a sit-in near Media City, often harassing TV personalities and forcing many of them to sneak into the studios from other entrances.
Demonstrators said they gathered there to protest the lack of local media coverage of their activities, and insisted their gathering was peaceful. Health ministry official Khaled el-Khateeb said 23 people were injured in the clashes; and security officials said two conscripts were also wounded, including one with birdshot.
The security officials, who spoke on condition of anonymity because they were not allowed to address the media, said 31 rioters were detained following the clashes. Footage of the detainees sitting on the ground outside the media city was aired on private channels.
The new unrest came as state-controlled TV reported that security forces will establish a cordon within 48 hours around the two main protest sites in Cairo where thousands have been camped out since before Morsi was ousted by the military on July 3.
The government offered protection and "safe passage" to those willing to leave the two main camps -- a large one outside the Rabaah al-Adawiya Mosque in eastern Cairo and a smaller one near Cairo University's main campus in Giza. The leadership had earlier given orders to police to end what it described as "threat to national security" and sources of "citizens' terrorism."
Authorities will let people leave without checking their identities or arresting them, but they will not allow anyone into the protest camps, the report said. It did not elaborate on the next steps, but the government earlier said it will use water cannons and tear gas in dispersing the crowds.
The Morsi supporters are also planning rallies late Friday outside security headquarters near one sit-in site, including the Republican Guard club, where they had staged a protest that turned bloody last month, and another army building.
The security cordon around the protest camps raises the possibility of new violence, which has killed more than 130 Morsi supporters and injured hundreds since the military coup. The ouster followed mass demonstrations calling for Morsi to step down after a year in office, saying his policies had failed and he had put power in the hands of his Islamist group.
Facing domestic and international pressure to avoid bloodshed, authorities have taken the unusual step of going into details of its security plans.
Interior Minister Mohammed Ibrahim told a newspaper that police have finalized plans for breaking up the sit-ins, and were awaiting orders from prosecutors to start the second phase of its operation.
Police have given authorities information about weapons in the protest camps and the "dangers emanating" from there, and that the next phase of the plan, which includes surrounding the sites, would begin within hours, he was quoted as saying.
"The forces have established their presence in various areas with the aim of protecting security and stability," Ibrahim said, adding that the ministry was awaiting legal action from the prosecutors. He said a prosecution team will accompany the security forces to monitor how they deal with the protesters.
Ibrahim told the newspaper that he was awaiting approval from the National Defense Council on measures relating to the final phase of the operation, which would be the use of force while trying not to injure anyone.
The Amnesty International report quoted a survivor of an attack by Morsi supporters near the Cairo University sit-in as saying that he saw one bloodied man have his throat cut and another stabbed to death.
The report also cited accounts from survivors that Morsi backers also abducted and tortured their political opponents with beatings and electric shocks at or near the protest sites.
The Interior Ministry last weekend had said 11 bodies were found near one of the protest sites, with some showing signs of torture, apparently by members of the sit-ins who believed the victims were spies.
Near the Rabaah protest camp, people armed with sticks and makeshift body armor stood guard behind walls of sandbags, tires and bricks.
One speaker defiantly told the crowd that the military leader, Gen. Abdel-Fatah el-Sissi, appeared reluctant to carry out his promise to break up the sit-ins.
"This man is about to fall now in the face of all these retractions," the speaker said to thousands of people who gathered for a meal to break their daytime fast for the holy month of Ramadan.
Ahmed Madani, 26, was installing a new tent at one of the camp entrances, saying the facilities will have a kitchen and toilets.
"We are here to show them that we are determined to stay and we won't give up," he said. "Even if I have to die, I will not leave. We are thousands ready to die for our cause."
The pro-Morsi camps have disrupted daily life in Cairo, blocking traffic and antagonizing some residents already suffering under Egypt's economic woes.
"A peaceful sit-in does not block roads, it doesn't terrorize people, it does not kill people and it does not attack people," said Wahid Idris, an opponent of the Muslim Brotherhood. "I want them to use any means to put an end to that sit-in."
In addition to the smaller sit-in across town, a new vigil sprung up briefly near Cairo's international airport, on the outskirts of the suburb of Heliopolis, in a neighborhood known as "The Thousand Houses."
An Associated Press reporter saw thousands of protesters, many of them are families and women in conservative Muslim dress, occupying a square and laying prayer rugs on the asphalt. They raised banners with Morsi's portrait saying, "Down with military rule," waved Egyptian flags and chanted, "Go away Sissi! Morsi is my president."
About six hours later, however, the camp was dismantled because organizers believed it to be insecure, said Adel Hassan, a protester who folded his tent.
U.S. Deputy Secretary of State William Burns arrived in Cairo, and an Egyptian Foreign Ministry official said he would meet with interim leadership officials and representatives from the Muslim Brotherhood and its allies Saturday.
Amr Darag, one of the Brotherhood negotiators who will meet with Burns, told the AP that the group and its allies are looking for "confidence-building measures" in order for them to sit at the table with their rivals.
Such measures include releasing detained Brotherhood leaders, unfreezing the group's assets, lifting the ban on its TV stations and ending violence against its protests. Darag said the group can't order its protesters to go home because they are fighting for their rights and the reinstatement of Morsi as president.
It was unclear if Burns would see Morsi during his second visit to Cairo since the coup. On Monday, top European Union diplomat Catherine Ashton saw Morsi for two hours at the facility where he is being held by the military. An African Union delegation also briefly met the ousted president a day later.
In London, Kerry sought to clarify controversial remarks he made Thursday about the crisis when he told Geo TV in Pakistan that the Egyptian military was "restoring democracy."
The comment was seen by some as a signal the U.S. was siding with the military, even though the State Department has repeatedly said the U.S. is not taking sides.
Kerry said Friday that all parties -- the military as well as the Morsi supporters -- should be inclusive and work toward a peaceful resolution of the crisis.
"The last thing that we want is more violence," he said. "The temporary government has a responsibility with respect to demonstrators to give them the space to be able to demonstrate in peace. But at the same time, the demonstrators have a responsibility not to stop everything from proceeding in Egypt."
A spokesman of Egypt's Mulim Brotherhood, Gehad el-Haddad, denounced Kerry's remarks, asking if Kerry would similarly approve of Defense Secretary Chuck Hagel deposing President Barack Obama if large protests took place in the United States.
Rights groups have warned against using force to end the protests. The New York-based Human Rights Watch urged the interim leadership to take all measures to avert bloodshed.
"To avoid another bloodbath, Egypt's civilian rulers need to ensure the ongoing right of protesters to assemble peacefully, and seek alternatives to a forcible dispersal of the crowds," said Nadim Houry, deputy director of the Middle East and North Africa division of Human Rights Watch.
The Muslim Brotherhood has opposed all measures taken by the military since the coup, including the appointment of an interim president, the suspension the constitution and the disbanding of the Islamist-dominated legislative council.
Associated Press Tony G. Gabriel contributed to this report.
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