Obama campaign manager made the offer in a letter sent Friday to Romney campaign manager Matt Rhoades, CNN reported.
The letter was posted on the Obama campaign Web site and e-mailed to journalists.
"If the governor will release five years of returns, I commit in turn that we will not criticize him for not releasing more -- neither in ads nor in other public communications or commentary for the rest of the campaign," Messina wrote.
Rhoades rejected the letter two hours later.
"It is clear that President Obama wants nothing more than to talk about Governor Romney's tax returns instead of the issues that matter to voters, like putting Americans back to work, fixing the economy and reining in spending," he replied.
The presumptive Republican presidential nominee told reporters Thursday he went back and looked at his taxes, "and over the past 10 years I never paid less than 13 percent.
"I think the most recent year is 13.6 or something like that. So I paid taxes every single year," he said, countering suggestions from Senate Majority Leader Harry Reid, D-Nev., that Romney, one of the richest Americans ever to win a major party's presidential nomination, didn't pay any taxes for a decade because of his success at taking advantage of tax breaks.
Reid didn't provide proof to support the assertion, but said it came from an investor at Bain Capital, the private equity firm Romney founded and ran.
Romney's campaign didn't release new details of his tax returns after Romney made his assertion about his tax rate, but he assailed Reid for his allegations.
"Harry Reid's charge is totally false," Romney said at a hastily arranged news conference in South Carolina, where he was on a fundraising campaign stop. "I'm sure waiting for Harry to put up who it was that told him what he says they told him. I don't believe it for a minute."
Romney added, "The fascination with taxes I've paid I find to be very small-minded compared to the broad issues that we face."
The Obama campaign responded to Romney's news conference by demanding he supply proof of the tax rate he paid.
"Since there is substantial reason to doubt his claims, we have a simple message for him: Prove it," Obama campaign spokeswoman Lis Smith told The Wall Street Journal.
Obama campaign spokesman Ben LaBolt later told reporters on a conference call Romney "has the ability to answer all of these questions by releasing several years of tax returns. He simply hasn't done that."
A Reid spokesman echoed the Obama campaign.
"We'll believe it when we see it," Adam Jentleson told The Washington Post. "Until Mitt Romney releases his tax returns, Americans will continue to wonder what he's hiding."
Jentleson and LaBolt said Romney should follow the lead of his father, George Romney, who was a candidate for the Republican nomination for president in 1968 and released 12 years' worth of tax returns. The elder Romney explained that any one year might just be a fluke.
Mitt Romney has released returns for 2010, showing he paid an effective federal income-tax rate of 13.9 percent on $21.7 million income, most of it from investments. He also released an estimate of his 2011 returns.
His campaign insisted Thursday Romney would make public no more than those two years of tax returns.
The discussion comes as Obama and other Democrats are pushing to raise taxes on wealthier taxpayers.
In 2011, Obama and first lady Michelle Obama reported an effective federal income tax rate of 20.5 percent. In 2010, their rate was slightly more than 26 percent.
Americans paid an average tax rate of 17.4 percent in 2009, a report from the Congressional Budget Office last month indicated.
The rate was the lowest average rate paid since 1979.
Notable deaths of 2014 [PHOTOS]
Cyclist struck and dragged underneath car