Senior Conservatives said there was no longer support in Parliament for a so-called "hard Brexit" after the party saw its Commons majority wiped out.
Former chancellor George Osborne predicted that Mrs May could be forced out of No 10 within a matter of days, saying she was "dead woman walking".
Graham Brady, the chairman of the Tory backbench 1922 Committee and one of the party's key power-brokers, insisted that there was no appetite among MPs for an immediate leadership challenge which could see them plunged into another general election.
However he acknowledged that the party would have to abandon much of the programme set out in the general election manifesto as it would no longer be able to get it through Parliament.
Earlier, Arlene Foster, the leader of Northern Ireland's Democratic Unionists (DUP), confirmed she would be meeting Mrs May in Downing Street on Tuesday with a view to finalising a deal to prop up a minority Tory government.
She said they had had "very good discussions" in Belfast on Saturday with Conservative chief whip Gavin Williamson, but declined to be drawn on the details.
While the DUP campaigned to leave the EU in last year's referendum, it has refused to endorse Mrs May's position that "no deal is better than a bad deal" - insisting that there must be no return of the "hard border" with the Republic.
At the same time Ruth Davidson, leader of the Conservatives in Scotland - where the party reversed the trend in England and gained MPs - has called for a new approach building cross-party support for an "open Brexit".
Chancellor Philip Hammond, meanwhile, is reported to have told the Prime Minister that she needed to put "jobs first" in negotiating a new deal with Brussels, in comments seen as a coded attack on her focus on controlling immigration.
Mr Osborne, who was sacked by Mrs May and now is editor of the London Evening Standard, said there was now no majority in the Commons for a "hard Brexit".
"The DUP need a deal because they are absolutely committed not to have a hard border with the Republic of Ireland," he told BBC One's The Andrew Marr Show.
"Theresa May's central claim which is no deal is better than a bad deal now becomes undeliverable because the DUP will never allow no deal."
Defence Secretary Sir Michael Fallon said senior ministers had made clear to Mrs May she operated in the light of the election result.
"We are going to see, I hope, more collective decision-making in the cabinet. I and other senior colleagues have made that clear to her," he told The Andrew Marr Show.
"I think you will also see that she will want to work much more closely with the parliamentary party both in the conduct of business and the development of policy."
While he played down suggestions that would involve rethinking the approach to Brexit, he indicated that there would be a new emphasis on business.
"We want to work with business on this. It is very, very important that we are careful about the existing trade we do with Europe, about access to the single market in whatever new arrangement that we come to," he said.
Sir Michael did not rule out the possibility of co-operating with the Labour Party over the negotiations.
"I welcome the fact the Labour Party have shifted their view now and do not seem to be calling for a new referendum," he said.
"They, like us, I hope now want to have a successful Brexit, an agreement that works for us, that works for the European Union, that does not jeopardise the jobs and trade we do with Europe, but still implements the overall result of the referendum last year."
Labour leader Jeremy Corbyn, appearing on ITV's Peston on Sunday, confirmed that the party - like Mrs May - was committed to leaving the EU and pulling out of the single market, while seeking a "jobs-first Brexit".
"Where I frame it is, we want a tariff-free access to the European market, we also want to maintain a very important university and research collaboration in Europe," he said.