"I think at this point of time it's going to dog him all the way and he needs to get it behind him," party strategist Ed Rollins said on "Fox News Sunday."
"I think he needs to release more taxes. Absolutely."
Rollins managed the presidential campaigns of former Arkansas governor Mike Huckabee, former president Ronald Reagan and, most recently, Michele Bachmann, the Minnesota congresswoman he recently eviscerated publicly for her attacks on an aide to Secretary of State Hillary Clinton.
The veteran Republican said he would have advised releasing the tax returns years ago had he been on Romney's campaign team, although he added that claims from Harry Reid, the Senate majority leader, would have nothing to do with that decision.
Reid said last week that a "very reliable" source, a former Bain Capital investor, has told him that Romney, a multimillionaire whose income is derived mostly from investments, didn't pay taxes for 10 years.
Romney has vehemently denied that claim, and suggested the White House has concocted a cunning scheme, with Reid in the starring role, aimed at forcing him to release more of his returns.
"I would not put out 20 years and I obviously wouldn't respond to anything Harry Reid states," Rollins said.
"At the end of the day you come to the point where you basically give a little bit more and you move forward. And he's going to do that. Two years is not enough, obviously."
Democrats have pounced on Romney's refusal to release more tax returns and are showing no signs of letting up, given polls are on their side. Most surveys suggest the majority of Americans, including the all-important bloc of independent voters, want to see more of Romney's tax information.
President Barack Obama's top strategists made the rounds of the crucial Sunday morning talk show circuit, keeping up the pressure. Robert Gibbs, the former White House press secretary who's now on Obama's Chicago-based re-election team, urged Romney to "go to Kinko's."
If Romney wants to shut down Reid, Gibbs said, all he has to do is release his tax returns and hand out copies to the media, Reid and other interested parties.
"We can put all of this to rest tomorrow," he said on CNN's "State of The Union."
"Mitt Romney can go to Kinko's, he can photocopy his tax returns — they're several hundred pages. He can hand them out to people like CNN, he can hand them out to reporters all over the country, and you know what? We wouldn't talk about this tomorrow. The whole world would know exactly what loopholes he's taking advantage of."
David Axelrod, Obama's top adviser, joined in.
"They gave 23 years of tax returns to the John McCain campaign, they've given one year of tax returns to the American people," he said on Fox News, referring to Romney's team.
"It was Gov. Romney's father who pioneered the release of tax returns when he ran for president because he said one year can be misleading. Why don't they just put this to rest? Why is he hiding?"
Gibbs was called upon repeatedly to answer Romney's claims that the White House was behind Reid's attacks. He stopped short of a denial, saying only: "I don't think anybody controls Harry Reid."
As for whether Team Obama would direct Reid to ease up, he added: "I think it's important that we know the financial backgrounds of candidates. We're going to have a pretty big debate over tax reform in the next few months and we have to know what's in people's tax returns and what they're getting the benefit for."
But there were Republican attack dogs out in full force on Sunday as well, including Senator Lindsay Graham, who branded Reid a liar on CNN.
Graham charged that Reid "made up" the accusations against Romney, levelled in the U.S. Senate last week.
"What (Reid) did on the floor was so out of bounds; I think he's lying about his statement, knowing something about Romney," he said.
"I just can't let that pass, I just cannot believe that the majority leader of the United States Senate would take the floor twice, make accusations that are absolutely unfounded in my view, and quite frankly making things up to divert the campaign away from the real issues."
Reince Priebus, chairman of the Republican National Committee, called Reid a "dirty liar" on ABC's "This Week."
"I'm not going to respond to a dirty liar who hasn't filed a single page of tax returns himself," Priebus said, alleging that Reid "complains about people with money but lives in the Ritz Carlton here down the street."
"So if that's on the agenda, I'm not going to go there. This is just a made-up issue. And the fact that we're going to spend any time talking about it is ridiculous.”
Other Republicans have agreed with Rollins, however, calling on Romney to do as his father, George, did in 1967 during his own run for president: he released 12 years of his tax returns, and is credited with pioneering such tax return transparency among politicians.
Those Republicans include former Mississippi governor Haley Barbour, conservative columnist George Will and Bill Kristol, editor of the Weekly Standard.