Blair, who is said to have close links with the UK tabloid's owner Rupert Murdoch, wanted Brown to get his ally Tom Watson to lay off the News International title, but Brown refused, the 'Daily Mail' quoted sources as saying.
Watson's two-year crusade played a major part in Murdoch's shock decision to close the newspaper after Sunday's edition, the report said.
Brown's office has declined to comment on whether Blair had intervened, saying, "We never comment on private conversations."
But a friend of Brown said: "There is no doubt about it, Tony wanted Gordon to intervene."
A spokesman for Blair, however, said: "The allegation is categorically untrue." He declined to elaborate on which aspect was untrue.
Blair's alleged attempt to persuade Brown to put pressure on Watson is likely to bring his links with Murdoch under fresh scrutiny.
On becoming Labour leader in 1994, Blair flew to a conference hosted by Murdoch to end Labour's feud with him. The alliance continued throughout Blair's 10 years as British Prime Minister at 10 Downing Street, as Murdoch's papers, The Times, The Sun and the News of the World supported him.