Embargoed to 2230 Friday August 23
It is understood that the PM put forward the former chancellor as a credible candidate to replace Christine Lagarde in a phone call with Trump this week.
And Mr Johnson is expected to continue pressing the case of his former Conservative colleague, currently editor of the Evening Standard, when he meets with leaders of the world’s richest countries at the G7 summit in Biarritz over the weekend.
The move comes after Mr Osborne, who had previously clashed with Johnson over Brexit, threw the Standard’s weight behind the new PM in the race to succeed Theresa May as Conservative leader.
Ms Lagarde will step down as IMF managing director after eight years on 12 September to take up the post of director of the European Central Bank and the international body wants to reach a decision on her successor by October 4.
Friends of Mr Osborne - often pilloried for his multiple jobs - have let it be known that he is giving “serious thought” to putting his name forward for the presitigious IMF post.
But the UK missed an opportunity to nominate him earlier this month when Bulgarian Kristalina Georgieva, chief executive of the World Bank, was chosen by European governments as their candidate.
By convention, the IMF is always led by a European national, while a US citizen takes the presidency of the World Bank.
Ms Georgieva’s victory over rivals including former Eurogroup president Jeroen Dijsselbloem and ex-EU commissioner Olli Rehn in the battle for the EU’s nomination does not guarantee her the IMF job. But Mr Osborne would need strong support from non-EU economies including the US to have any chance of supplanting her, and he is regarded as a long-shot in the race for the Washington-based post.
The 66-year-old Bulgarian this week cleared a further obstacle in her way when the IMF’s executive board scrapped a longstanding age limit of 65 for a new managing director.
Mr Johnson’s friendship with Mr Osborne, which goes back to their time together as students at Oxford University, was strained by the 2016 EU referendum in which the pair - both tipped as future prime ministers - were combative figures on opposing sides.
At the time, Osborne suggested that Johnson’s support for Leave may be motivated by personal ambition, while the Brexit figurehead dismissed the warnings from David Cameron’s right-hand man about the dangers of EU withdrawal as “Project Fear”.
As recently as May this year, Mr Osborne raised concerns about Mr Johnson’s character, saying there had “always been two Borises”, the hard Brexit politician and the liberal internationalist London mayor.
However, in June he wrote in the Standard that he was backing Johnson as “the candidate who has a chance of uniting this divided government”.