Republican Party chairman Reince Priebus called Reid a "dirty liar." Sen. Lindsey Graham, R-S.C., said, "I think he's lying about his statement of knowing something about Romney" and he contended Reid was "making things up to divert the campaign away from the real issues."
Reid's spokesman, Adam Jentleson, contended that Republicans were trying to cover up for Romney and the aide repeated the Nevada senator's claims that the source about Romney's taxes is "credible." Other Democrats said Romney could clear up any confusion about his taxes by disclosing them to the public, and the issue was a main talking point on the Sunday news shows.
Reid said last week he had learned of the claim about Romney's taxes from an investor in Romney's former firm, Bain Capital. But the lawmaker hasn't offered any evidence to back it up and even has said he's not sure it's true.
Romney, who has disclosed that he's worth as much as $250 million, has said Reid's claim isn't true and that he's paid "a lot of taxes."
Romney's 2010 federal tax return shows he paid 13.9 percent tax on income of $21.6 million. Most of Romney's income came from investment gains, which are taxed at a lower rate than earned income.
The candidate has refused to release more than one year of personal tax returns, despite calls from Democrats and some Republicans to do so, saying his critics would distort the information and use it against him. He has promised to release a second year of returns.
Democrats have tried to make Romney's personal wealth and how he's managed it a key issue in the presidential contest. The former Massachusetts governor, who would be among the richest presidents ever elected, is aggressively competing with President Barack Obama for the support of middle-class voters.
Democrats pressed Romney to providing more tax information.
"He could clear it up just like that, lickety-split, by releasing his tax returns, which every major candidate for president of the United States has done except for Mitt Romney," said Rep. Debbie Wasserman Schultz of Florida, who heads the Democratic National Committee.
Jentleson, Reid's spokesman, said in an email that "it's clear Mitt Romney is hiding something, and the only way for him to clear this up is to be straight with the American people and release his tax returns."
Graham appeared on CNN's "State of the Union," while Priebus and Wasserman Schultz were on "This Week" on ABC.
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