The former prime minister has met with Jared Kushner three times since September to discuss politics, with the latest meeting last week taking place within the West Wing, according to the paper.
But a spokesperson for the Tony Blair Institute this morning said: “The story in the Mail on Sunday is an invention.”
Last night, Blair’s official spokesperson refused to answer questions around the reports, telling the Press Association: “I’m not going to comment on private conversations.”
But now the Institute hit back: “Mr Blair has made no such ‘pitch’ to be the President‘s Middle East envoy.
“Neither has he had any discussions about taking such a role or any role working for the new President.
“He has been working on the peace process for 10 years. He continues to do so. He does so in a private capacity. He will continue to do it in that way. Period.”
Blair, who led the Labour Party to three general election victories, has specialised in Middle East policy since stepping down as prime minister in 2007.
He took the role of Middle East envoy for the Quartet Group comprising the EU, US, Russia and UN.
His expertise would likely be seen as highly useful to the fledgling and inexperienced Trump administration.
In his role as advisor to the President, Kushner, who is married to Trump’s daughter Ivanka, has taken a lead on foreign policy.
Blair and Kushner are reported to have first met to discuss politics in September last year and once more after Trump’s victory in November.
It comes after the Trump administration set its sights on bringing about a conclusion to the decades-old Israel-Palestinian conflict.
Last month, Trump rejected the long-established US policy of advocating a two-state solution to the conflict.
Instead, he said, both sides would have to make compromises - and he was intent on making “the ultimate deal”.