Republican presidential candidate Mitt Romney said Thursday that he has paid a federal income tax rate of at least 13 percent in each of the past 10 years, bowing to months of political pressure to disclose more information about his vast personal fortune.

“I did go back and look at my 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 told reporters here Thursday.

President Obama and his allies have been hammering Romney relentlessly, both in television advertisements and on the campaign trail, for refusing to disclose more than two years’ worth of tax returns. They have suggested that Romney, one of the richest Americans ever to win a major party’s presidential nomination, is hiding something about his personal finances.

Romney’s disclosure Thursday came in response to a reporter’s question at a news conference that Romney had hoped would focus on a white-board presentation he gave outlining his latest attack on Obama over Medicare. Romney’s advisers insisted that the former Massachusetts governor had not planned to provide new information about his taxes but that, when asked about his tax rate, he provided an honest answer.

Romney sounded rankled by the very question, though, saying, “the fascination with taxes I’ve paid I find to be very small-minded compared to the broad issues that we face.”

If the candidate hoped his answer would put the issue to rest, he ended up only inviting more questions. Obama campaign spokesman Ben LaBolt quickly challenged Romney to “prove it.”

“He has the ability to answer all of these questions by releasing several years of tax returns. He simply hasn’t done that,” LaBolt told reporters on a conference call. LaBolt noted that Romney’s father, George, released 12 years of income tax returns when he ran for the Republican presidential nomination in 1968.

In a letter to the Romney campaign Friday, Obama campaign manager Jim Messina challenged the Republican candidate to release just five years of tax returns, pledging that the Obama campaign would then stop criticizing him for not making public more of his returns. Messina said the request for Romney’s “complete returns” filed between 2007 and 2012 “is surely not unreasonable.” He said it would require the release of only three more sets of returns in addition to the 2010 tax-year return that Romney has already released and the 2011 return he has pledged to provide this year.

The additional returns would “help answer outstanding questions raised by the one return he has released to date, such as the range in the effective rates paid, the foreign accounts maintained, the foreign investments made, and the types of tax shelters used,” Messina wrote.

In 2011, Obama paid an effective rate of 20.5 percent, according to his tax returns. Americans paid an average tax rate of 17.4 percent in 2009, according to a report from the Congressional Budget Office that was released last month — the lowest average rate paid since 1979 and a drop from the 19.9 percent paid in 2007.

There is no legal requirement for presidential candidates to release any income tax returns, although the Federal Election Commission requires candidates to release personal financial disclosures.

Romney’s disclosure estimates his personal fortune at between $190 million and $250 million.

Romney has long resisted providing more than two years’ returns because he said his Democratic opponents would use the details of his financial holdings as the basis of campaign attacks.

In an interview with NBC’s “Rock Center” that aired Thursday, the candidate’s wife said Democrats want the records for “ammunition.” “The more we get attacked, the more we get questioned, the more we get pushed,” Ann Romney said, adding that the couple has been “very transparent to what’s legally required of us.”

“Mitt is honest,” she said. “His integrity is just golden.”

In January, under pressure from his Republican primary rivals, Romney released his 2010 tax return, which showed he paid 13.9 percent on $21.7 million in 2010 income. That is the figure he was referring to Thursday, his campaign said, when he mentioned a 13.6 percent tax rate. He has pledged to release his 2011 filings as soon as they are completed later this summer or fall.

A Romney spokesman confirmed that in saying he paid a rate of at least 13 percent in each of the past 10 years, Romney was referring only to his federal income taxes and not personal property or sales taxes.

The question to Romney at Thursday’s news conference was a follow-up to a response the candidate gave ABC News’ David Muir in an interview last month. Muir asked whether Romney had ever paid less than the 13.9 percent he paid last year.

“I haven’t calculated that,” Romney told Muir. “I’m happy to go back and look, but my view is I’ve paid all the taxes required by law.”

In recent weeks, Romney’s campaign has been beating back accusations from Senate Majority Leader Harry M. Reid (D-Nev.), who has claimed that he was told by an investor at Bain Capital, the private equity firm Romney founded and ran, that the Republican candidate had paid nothing in taxes for at least 10 years because of his success at taking advantage of tax breaks.

Romney on Thursday dismissed Reid’s charge as “totally false.” “I’m sure waiting for Harry to put up who it was that told him what he says they told him,” Romney said. “I don’t believe it for a minute, by the way.”

A spokesman for Reid said Romney could prove his statements true by making his records public. “We’ll believe it when we see it,” said the spokesman, Adam Jentleson. “Until Mitt Romney releases his tax returns, Americans will continue to wonder what he’s hiding.”

Democrats have tried to present Romney’s unwillingness to release more filings as an illustration of their argument that his tax policies would favor the wealthy, like himself.

Romney pays a lower effective tax rate than many middle-class families. His overall rate is so low in part because most of his income comes from investments, and investment income is taxed at a lower rate than salaries and wages.

His investment income is taxed at 15 percent. Being in the highest tax bracket, Romney would pay a marginal rate of 35 percent if all his income was regular wages.

At his news conference, Romney suggested that his charitable contributions should also be taken into account. “Every year, I’ve paid at least 13 percent, and if you add, in addition, the amount that goes to charity, why the number gets well above 20 percent,” Romney said.

Helderman reported from Washington. David Nakamura contributed to this report.