Mr Osborne was said to have been angered by Mrs May's low-key support for the Remain campaign in last year's EU referendum.
According to Theresa May: The Enigmatic Prime Minister by Rosa Prince, serialised in the Daily Mail, the then-chancellor was so frustrated by her "submarine" tactics, he told a number of allies she could end up being demoted.
However, it was Mr Osborne who lost out when Mrs May finally made it to No 10 when one of her first acts as Prime Minister was to sack him.
Former cabinet minister Sir Eric Pickles, who was also formerly the leader of Bradford Council, said: "We had that ridiculous thing where Osborne was going to have her sacked because she wasn't doing enough. By God, she put him in his place."
Mr Osborne was said to have been hoping to carry on in a senior role when he was summoned to Downing Street, despite previous disagreements with Mrs May.
An ally is quoted as saying: "He wanted the Foreign Office. There was some speculation that he could even stay on at the Treasury - he'd have done that."
Instead, he was told that there would be no place for him at the Cabinet table in a conversation that lasted just 10 minutes.
Afterwards, the book said, Mrs May's aides were almost certainly responsible for letting it be known that she had given him a severe dressing down, telling him he needed to show more humility if he ever wanted to be prime minister.
His dismissal was said to have followed a number of earlier clashes, mainly over immigration, with the then-chancellor arguing for a more relaxed approach to economic migration.
At one cabinet meeting in 2012, Mr Osborne was said to have "declared war", describing the "horror story" of a Chinese businessman with millions to spend on British products who had been stopped at Heathrow, where he was interrogated for hours and subjected to a full strip search.
He said on his release, the man had returned on the next flight to China where he told his friends to avoid the UK and demanded to know who was responsible for this "debacle".
Mrs May, whose department was responsible for the Border Agency, was said never to have forgiven him. One minister in her current Cabinet said: "She couldn't stand him after that."
Mr Osborne was not the only minister with whom she was said to have fallen out. Michael Gove was another to pay the price for a long-term feud.
Mrs May, who blamed him for a bitter clash over Islamism in schools that led to the sacking of one of her closest aides, dismissed him as justice secretary in a conversation lasting just two minutes.