The bitter rivalry between the two most powerful figures in the Conservative Party, David Cameron and Boris Johnson, is laid bare today in an updated book on the rise of the Tory leader.

One associate of both men says that Mr Johnson, the Mayor of London, despises Mr Cameron, does not respect his intelligence and cannot understand how he got the top job. Mr Cameron, for his part, sees Mr Johnson as being dangerously spirited and lacking the necessary moral cutout or alarm system to be a serious politician.

The book, Cameron: The Rise of the New Conservative, by Francis Elliott, deputy political editor of The Times, and James Hanning, reveals that Mr Cameron regarded the idea of the former Editor of the Spectator becoming Tory choice for mayor as outlandish and agreed to it as a last resort.

Even then the Tory leader could not bring himself to make the approach to the man whom he had refused to promote from a lowly front-bench job. For his part Mr Johnson was reluctant to accept the role and kept Mr Cameron sweating as emissaries were sent to persuade him.

The revelations shed new light on the tensions between the two men as an exclusive poll of grass roots found Mr Johnson has a higher approval rating than Mr Cameron. The poll of 1,368 Tory members by the website ConservativeHome gave Mr Johnson a net satisfaction rating of 86 per cent, against 84 per cent for Mr Cameron.

This week Mr Johnson risked reigniting the most damaging row that Mr Cameron has seen since taking over, by praising school selection by academic ability. Mr Johnson dismissed policies designed to give parents more choice and control over schools, a key element of Conservative education policy, and said: “Choice will be pretty meaningless until you bring back academic selection.”

The comments go to the core of the dispute over grammar schools that caused the biggest grassroots revolt of Mr Cameron’s leadership. According to the ConservativeHome poll more than 55 per cent of members strongly disagree with Mr Cameron’s opposition to new grammar schools.

The Tory leader was reported yesterday to have strengthened his party’s opposition to the new 50p tax after Mr Johnson came out unambiguously against it. Aides insisted that the policy had not changed. Nevertheless Mr Johnson’s forthright remarks on education and tax are the latest of a series of policy differences where he has appeared keen to set himself on a different course from Mr Cameron, who was a contemporary at Eton and Oxford. The two men were always regarded as rivals rather than friends.

Mr Johnson appeared deliberately to leave open the idea that he might one day be a threat to Mr Cameron by disclosing on Thursday that he might not stand for a second term as mayor, further fuelling the idea that he sees himself as a future prime minister.

Asked by the London Evening Standard whether he would be a one-term mayor he replied: “Nothing is excluded.”

He has denied, however, that there is any enmity between him and Mr Cameron. “I’ve always been very fond of David Cameron and held him in extremely high regard,” he said. “I was one of the first people to urge him to stand for the party leadership. I regard him as a friend. He will be an excellent prime minister and any suggestion otherwise is tripe.”

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