Emails released under freedom of information laws showed the Department of Health under Labour tried to stop details emerging of a hospital scandal involving higher-than-normal death rates.
The then Labour health secretary was recorded as being "furious" when news leaked of poor standards of care in November 2009, months before the General Election.
His successor Mr Hunt seized on the emails as "shocking revelations on Andy Burnham's attempts to cover up failing hospitals".
"We're legislating to make sure this can never happen again," he wrote on Twitter.
However, Mr Burnham insisted the correspondence showed only that he was concerned disclosure rules had been broken, and challenged Mr Hunt to either substantiate his claims or apologise.
"It is impossible to see how you can claim this mounted to a cover-up," he told him in a letter.
"I therefore ask you, by the end of today, to provide me with evidence to substantiate your assertion.
"If you fail to provide such evidence, I will require a full retraction and public apology. If that is not forthcoming I will consider further action."