Mr Johnson, one of the Conservatives' most popular politicians, tweeted that an article in the Mail on Sunday newspaper headlined "Boris set to launch bid to be PM as May clings on" was "tripe".
Having just returned from a meeting with the Queen to discuss the formation of a new government Theresa May said 'We will continue to work with our friends and allies, in the Democratic Unionist Party in particular. "Let's get on with the job", he said.
British media have reported that moves were afoot within May's party to dislodge her after her election gamble - aimed at increasing her party's majority in parliament ahead of Brexit talks - backfired.A leadership contest can be triggered if 15 percent of the Conservative's 318 lawmakers write to Brady saying they no longer have confidence in May. The negotiations had been due to start around June 19. Labour surpassed expectations by winning 262.
Meanwhile, Jeremy Corbyn, the main opposition Labour Party leader who managed to win 30 more seats for the party, said, "I can still be prime minister". To begin with, it was understandable that ##theresa May stayed on because stability is vital, however, as days have passed her leadership looks more desperate than stable.
"The new cabinet obviously will meet early next week, our view of Brexit I don't think has changed", Defence Secretary Michael Fallon told the BBC, adding he believed the government would be able to muster parliamentary support for its Brexit plans.
But Graham Brady, who chairs the influential 1922 Committee of backbench Conservative lawmakers, said a "self-indulgent" party leadership campaign would only cause more uncertainty.
In her post-election reshuffle, May said she appointed ministers that reflect "the wealth of talent and experience across the Conservative Party". "The votes aren't there now in the (House of) Commons for taking Britain out of all the economic arrangements we have with the European Union", he said.
Scottish Tory leader Ruth Davidson has spoken to Mrs May about her concerns regarding LGBT rights.
Can still be next British PM: Labour leader Corbyn
He said there was a "possibility" of voting down the Queen's Speech and Labour were going to "push all the way" to achieve that. He said the Brexit content of that amendment would be about negotiating as quickly as possible a "jobs-first Brexit".
Mrs May is looking to secure the DUP's support.
The future of the proposed alliance had already been thrown into confusion late Saturday after May's office announced that an outline agreement had been struck, only to backtrack and say that talks were still ongoing.
The DUP and Irish nationalists Sinn Fein are due to restart talks to form a new power-sharing government in Northern Ireland and avoid devolved power reverting to the British parliament in London for the first time in a decade.
In a television interview Sunday with Sky News at Downing Street, May say she would help deliver the "stability and certainty" the country needs during the reshuffling.
The deal with the DUP in Northern Ireland is on a "confidence and supply" basis, which means the party would lend its support to block no confidence votes and pass budgets.
The DUP is similar to the "religious right" in the United States and takes a hardline stance on social issues, such as same-sex marriage and abortion.
His comments come as the DUP confirmed that its party leader Arlene Foster is to hold talks with Theresa May in Downing Street to finalise an agreement on propping up her minority government.
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