Priebke died Friday in Rome, aged 100, where he was serving house arrest for a life sentence he was handed in 1998 over his participation in the 1944 Fosse Ardeatine mass killings, in which German soldiers shot dead 335 civilians.
A spokesman for the Catholic Church in Rome ruled out the possibility of Priebke's funeral being celebrated on Tuesday in a church in the center of the city, as had been suggested by his lawyer Paolo Giachini.
"No funeral rites are planned," Father Walter Insero said, as quoted by the ANSA news agency.
On Friday, Giachini had said that Priebke wanted to be buried next to his wife in the Argentine town of Bariloche. He was extradited from there in 1995, after a U.S. TV crew discovered his whereabouts and interviewed him.
But the foreign ministry in Buenos Aires closed that option. "Foreign Minister Hector Timerman has given the order not to accept the slightest move to allow the return of the body of Nazi criminal Erich Priebke to our country," it wrote on Twitter.
"Argentines will not accept this kind of affront to human dignity," the message said.
In Rome, Police Chief Fulvio della Rocca said he would ban any "form of solemn and public" funeral. He was backed by Mayor Ignazio Marino, who said that Rome was "an anti-Nazi and anti-Fascist city."
Marino said any ceremony "for someone who has inflicted so much pain on the people of this city" would have been inappropriate in the run up to the 60th anniversary of the deportation of Jews from Rome's ghetto, which started on October 16, 1943.
The dead at Fosse Ardeatine, a cave on the outskirts of Rome, included 75 Jews and at least one child. The massacre was thought to have been ordered by Adolf Hitler in response to the killing by Italian Resistance fighters of 33 German soldiers a day earlier.
In 2003, Priebke described it as "a personal tragedy" but did not ask for forgiveness. On other occasions, he repeated that he was simply following orders.
"Why don't they send him back to Germany? Isn't it enough all that he has said? I don't know if the Jewish community will want to do something, but we definitely don't want him here in Rome," said 87-year-old Giulia Spizzichino, who lost 26 relatives to the Nazis.
Marino said that Priebke would likely be given a private funeral in his local church, in Aurelio. He also said he was verifying whether he could legally stop the war criminal from being buried in his city.
"The risk is that he could be given a tomb in Rome and that it could become a pilgrimage site. Like it has been for (former Italian Fascist dictator Benito) Mussolini," Spizzichino said.
Giachini defended Priebke, praising his "courage, coherence and loyalty." A bunch of flowers was laid outside the late Nazi officer's Roman home, while "Honor to Priebke" and a swastika were scrawled on a nearby wall.
Priebke remained loyal to Nazi ideology until the end.
In a statement Giachini publicized after his death, Priebke said that "gas chambers were never found in (Nazi) concentration camps, except for the one built by the Americans when the war finished in Dachau."
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