Jim Messina wrote to Romney campaign asking for five years' worth of tax returns in exchange for an end to the issue

The Obama campaign team has kept up the pressure on Mitt Romney's tricky position on tax, promising to stop hounding him on the subject if he releases five years' worth of returns.

Obama's campaign manager, Jim Messina, wrote to his opposite number in the Romney camp, Matt Rhoades, making the offer in writing. The Romney campaign did not take up the offer.

The letter is aimed at addressing one of the frequent complaints by the Romney campaign that if he releases five years' worth of tax returns, the Obama campaign will demand 10, and if he releases 10, they will demand 20.

The Messina letter came after Romney, talking to reporters on the campaign trail Thursday, disclosed that he had paid at least 13.6% in taxes over a 10- year period. The Democratic Senate leader Harry Reid had claimed, based on an alleged informant inside Bain Capital, where Romney made his fortune, that the Republican presidential challenger had paid no tax whatsoever.

Reid, not satisfied with Romney's answer, called on his to release his tax returns to prove his 13.6% claim.

Romney has released his tax returns for 2010 and 2011 but the Obama team said he should follow the precedent of other presidential challengers by disclosing tax returns for a much longer period.

Messina, in his letter, wrote: "I am writing to ask again that the governor release multiple years of tax returns, but also to make an offer that should address his concerns about the additional disclosures.

"Governor Romney apparently fears that the more he offers, the more our campaign will demand that he provide. So I am prepared to provide assurances on just that point: if the governor will release five years of returns, I commit in turn that we will not criticise him for not releasing more – neither in ads nor in other public communications or commentary for the rest of the campaign."

Messina added that, given Romney has already released tax returns for 2010 and 2011, he would only have to release a further three, dating back to 2007.

Rhoades, in his reply, said: "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.

"If Governor Romney's tax returns are the core message of your campaign, there will be ample time for President Obama to discuss them over the next 81 days."

Romney's vice-presidential running mate, Paul Ryan, on Friday released tax returns showing he paid an effective rate of 20% in 2011, roughly in line with Obama's rate and likely higher than that of Romney. Ryan's release aligns him with Romney's position that giving out two years of tax returns is adequate.