White House spokesman Josh Earnest said Friday the release of the tax information is the kind of transparency voters think is important and is a "relevant part of the debate," The Hill reported.
Romney said this week he paid at least 13 percent in taxes every year for the past decade, the Los Angeles Times reported.
His rate was 13.9 percent in 2010 and 15.6 percent in 2011 on more than $20 million in income.
President Obama's campaign manager Jim Messina said Friday the campaign would stop asking for more tax records if Romney releases five years of his returns.
"Gov. Romney apparently fears that the more he offers, the more our campaign will demand that he provide," Messina wrote in a memo to Romney campaign manager Matt Rhoades.
"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 criticize him for not releasing more -- neither in ads nor in other public communications or commentary for the rest of the campaign," he wrote.
Rhoades wrote back to Messina, saying the tax-fight between the two candidates should not be at the forefront of the campaigns.
"It is clear that President Obama wants nothing more than to talk about Gov. 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 wrote.
Meanwhile, likely Republican vice presidential choice Paul Ryan released his tax returns for 2010 and 2011 Friday, showing he paid a higher rate than Romney.
Ryan, a U.S. congressman from Wisconsin, who filed joint returns with his wife, Janna, had an effective rate of 15.9 percent for 2010, when the couple reported income of $215,417, and 20 percent for 2011 on $323,416, the Los Angeles Times reported.
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