Scottish Conservatives leader Ruth Davidson last night ruled out standing, insisting she had an important job to do in Scotland.
Brexit campaign frontman Mr Johnson and the Remain-backing Home Secretary are seen as the two leading candidates to take the keys to Number 10. International Development Secretary Justine Greening said a leadership battle now would not be in the interests of the country and urged the pair to strike a deal to avoid a contest.
She said a “strong Conservative government” should have Mr Johnson at its centre to recognise the outcome of the EU referendum.
But Mrs May would also have to be at the heart of the administration as the country works on tackling issues around migration and freedom of movement.
However, both former leader Iain Duncan Smith and fellow Leave campaigner Liam Fox, who may stand, said the next leader should be pro-Brexit.
In an apparent warning to pro-Remain colleagues thinking of running for the leadership, Mr Duncan Smith told the BBC’s Andrew Marr Show that after voting for Brexit, the public would be unlikely to accept a new prime minister who wanted Britain to stay in the EU.
“Whoever takes up that job... it would be very, very difficult for the public who have voted for leaving the European Union to find that they then had a prime minister who actually was opposed to leaving the European Union,” he said.