HuffPost UK understands that the prime minister raised the issue during a phone call to the president, giving his full backing to the former Tory chancellor.
It is unclear if Trump was positive about the suggestion, and some government insiders caution that without European backing Osborne’s chances are doubtful.
However, the fact that Johnson personally raised the issue will inevitably spark suggestions of a significant warming of relations between the two Tory politicians, who have had radically different views of Brexit.
The former chancellor is now editor of the London Evening Standard, which gave a strong endorsement of Johnson’s bid for the Tory leadership this summer, before his landslide victory over rival Jeremy Hunt.
Osborne has not publicly said he is interested in the IMF job, which will be vacated by Christine Lagarde when she takes up her new role of the head of the European Central Bank.
But reports emerged last month that he had told friends he was open to the idea. If Osborne were to get the post, it would be the first time a Briton has headed the IMF in its 75 year history.
Osborne could face tough competition from Mark Carney, the current Bank of England governor whom he appointed.
However, European contenders for the job could also pip him to the post.