Mr Osborne, who was sacked by May and is now editor of the London Evening Standard, said there was now no majority in the Commons for a "hard Brexit", following Thursday's General Election.
He added: "I think we will know very shortly".
'She said I needed to get to know my party better. Boris Johnson is widely seen as her natural successor, but his public image was decimated following his role in last year's Brexit campaign.
"With only days to go before Brexit negotiations begin, the United Kingdom needs to be quick out of the blocks and agree on transitional arrangements, guarantee European Union citizens' rights and shift the focus of formal talks to future trading relationships".
We are going to see, I hope, more collective decision-making in Cabinet.
However, this was dealt an early blow after Downing Street mistakenly announced a deal had been reached with the Democratic Unionists to prop up Mrs May's minority Tory government.
Former Education Secretary Ms Morgan also predicted there could be a leadership challenge over the summer.
Nicky Morgan, another cabinet minister who was dumped by Theresa May, said: "I think it's fairly clear Theresa May can not lead us into another election - of course, we don't know when that's going to happen, and I don't think we should rush that".
"We are going to put down a substantial amendment to the Queen's speech which will be the main points of our manifesto so we will invite the House to consider all the issues we've put forward - jobs-first Brexit, policies for young people and on austerity", he told the BBC.
The Mail on Sunday claimed he had said in private that it is time to "go, go, go" on realising his long-held dream of becoming PM.
Whilst Emily Thornbury the shadow cabinet member said that May is "squatting in Downing Street".
She revealed a surprise comeback as environment secretary for Michael Gove - sacked after she beat him in last year's leadership battle.
Mr Brady said he did not believe there was any mood among Conservative MPs for a leadership contest that would create fresh instability.
"Certainly one of the things I was very keen to press home was that in the past the Conservative Party has been very poor in its communications with colleagues who lost their seats in the general election".