After weeks of talking about Romney's finances rather than President Barack Obama's jobs record, it is time, they say, for the multi-millionaire businessman to take the bat out of Obama's hand.
But the candidate is not for turning.
"We're going to put out two years of tax returns," Romney told the Fox & Friends programme on Sunday, referring to the 2010 return he has already released and the full 2011 return he has vowed to release before the November 6 election.
But there are some good reasons why Romney may continue with what to outsiders may seem like Sisyphean resistance.
Despite the heat, "you don't exactly see the bottom falling out of Romney's (polling) numbers", explained Bruce Haynes, a long-time Republican operative now with the bipartisan consultancy Purple Strategies.
According to Real Clear Politics' average of polls, Obama remains marginally ahead in the head-to-head race, but the gap has actually narrowed in the last few weeks despite the focus on Romney.
Given this, Haynes said, he would advise Romney not to release his returns, at least not yet.
"He just doesn't want to feed a machine that seems focused on his personal economic circumstances. That is not the narrative of the campaign that Romney would like to drive."
There seems little doubt that Romney releasing more returns would provide fodder for those arguing he is out of touch.
At the unlikely end of the scale, Romney could unwittingly provide evidence of legal wrongdoing that could shatter his White House bid.
More likely, it could reveal that Romney's effective tax rate is a fraction of that paid by most Americans, thanks to a series of offshore bank accounts and other tax loopholes.
"It could be that he maybe did something that was kind of shady, not in an illegal sense, but got his effective tax rate very low," said Martin Sullivan of Tax Analysts.
The structure of Romney's pay likely means that his remuneration from Bain Capital is taxed as a capital gain, at around 15%, rather than as an income tax at more than double the rate.
Romney's published return shows he paid an effective tax rate of 15% in 2010, by the pay being taxed as "deferred interest."
"Let's say he got his effective tax rate to5% or might even be 0%. That would be damning," said Sullivan. None of this would be illegal, but it would be "guilt by association."
"It is just putting more of a spotlight on his wealth, and reinforcing the general picture that the administration wants to convey that Romney, because he is rich, is out of touch."
There is no doubt that Romney is rich. According to David Friedman, chief of research firm Wealth-X, Romney is likely among the 3,140 richest individuals in the United States.
But, worryingly for the Romney campaign, in key swing states like Ohio nearly 50% of those polled believe he is too out of touch to be president.
Returns that show he is "not paying his share" would only fuel that sentiment.
The returns could also reveal details of Bain deals or other investment that are politically difficult to explain.
Already there has been speculation that he invested in firms that disposed of unborn fetuses, sent jobs overseas, sold arms to dodgy regimes and banks that helped fuel the financial crisis.
They could reveal that Romney, a devout Mormon did not tithe his 10% to the church.
Whatever the risks, the pressure is mounting on Romney to show his hand.
"This is what the Obama people have been praying for," former Republican-party chairman, Haley Barbour told the National Review. "He ought to release his returns."
But in taking the bat out of Obama's hand, will Romney only to replace it with a larger more punishing version?
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