Johnson had been a favorite to win the keys to Number 10 after Cameron quit, but he pulled out of the race.
May's party fell eight seats short of retaining its parliamentary majority, and is now in talks with Northern Ireland's ultra-conservative Democratic Unionist Party (DUP) which won 10 seats to forge an informal alliance.
There were warnings she would have to soften her stance on Brexit, as the DUP have been adamant that they can not afford to leave the European Union without a deal as it would mean a return to a "hard border" with the Republic.
While the to-and-fro between Downing Street and the DUP was unfolding, several British newspapers were reporting that some prominent Conservatives, including Foreign Secretary Boris Johnson and Brexit minister David Davis, were being urged by supporters to challenge May for the party leadership.
Robinson compounded the controversy when, in a later interview he said: "I wouldn't trust Muslims devoted to Sharia law, but I would trust them to go down to the shops for me". May tried to reassert her shattered authority at the weekend by announcing her new cabinet with no changes among her top team. "Let's get on with the job". Fallon said the DUP would agree to back the Conservatives on big economic and security issues.
"I don't detect any great appetite amongst my colleagues for presenting the public with a massive additional dose of uncertainty by getting involved in a self-indulgent Conservative Party internal election campaign", Graham Brady, the 1922 committee chairman, told BBC TV.
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Many critics, including Scottish Conservative leader Ruth Davidson, have expressed concerns over the DUP's stances on gay marriage and abortion, among other issues.
"There has been a lot of hyperbole about the DUP since Thursday, a lot of things said", DUP leader Foster told Sky News.
"I want an MP who answers to us - not to the Tories".
"But just to be clear, we will act in the national interest, we want to do what's right for the whole of the UK and to bring stability to the government of the United Kingdom". "With Theresa May weak, the hard-line Eurosceptics in the Conservative party, who are more organised than the Remainers, will be able to take the Prime Minister hostage in their pursuit of a hard Brexit", he added.
"The Prime Minister thanked Mr Kenny for helping to make UK-Ireland relations stronger than ever, wished him well for the future and said she looked forward to continuing a close relationship with his successor".
Party leader Arlene Foster later confirmed to the BBC that the Constitutional Research Council donated the money; party officials later said this explanation was enough and that they did not need to know the source of council's income.
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