Earlier, Mrs May announced that she meant to carry on in No 10 at the head of a minority government with the support of the Democratic Unionists (DUP) after falling eight seats short of an overall majority.
It was apparently sent on Sunday morning, following two days of speculation over whether Theresa May would be able to cling to power and if not, whether Mr Johnson is plotting to make a bid for the top job.
Downing Street said an outline agreement on a "confidence and supply" arrangement had been reached with the DUP which will be put to the Cabinet for discussion on Monday.
DUP leader Arlene Foster said her party would be entering talks with the Conservatives to explore "how it may it be possible to bring stability to our nation at this time of great challenge".
His last bid to lead the party, after David Cameron resigned in the wake of the Brexit referendum, ended before it began as he failed to secure support from senior colleagues such as Mr Gove.
Qatar diplomatic crisis: Amnesty bemoans human rights cost on Gulf nation's isolation
Still, the Pentagon has been developing contingency plans in case there's any interruption, defense officials said. Customers are seen shopping at the al-Meera market in the Qatari capital Doha , June 10, 2017.
"We need Bojo. We need a Brexiteer", they told the Mail on Sunday.
He said: "I am backing Theresa May".
Mrs May has insisted she will not quit immediately despite losing her Commons majority, but says she will "reflect" on the dire result amidst calls from pro-EU backbenchers for her to step down within months.
One ally told the Sunday Times: "We are facing a populist and they have realised we need someone who can talk to the people. Boris is the only option with the liberal values, Brexit credentials and popular appeal", he added.
"And move to a consensus within the country about what it means and what we seek to achieve as we leave".
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