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.
May called the election to try to strengthen her hand in Brexit talks, but her party lost its majority in parliament.
Instead, voters stripped the Conservatives of their parliamentary majority.
May's former director of communications, Katie Perrior, spoke openly Saturday about the behavior of the two aides, which she described as tyrannical, and resulted in an office that was "pretty dysfunctional".
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. But Johnson said he backed May.
May's aides, Nick Timothy and Fiona Hill quit on Saturday following sustained criticism within the party of the campaign.
Many of those Conservatives now opposing May are those who wanted to remain in the EU.
The change was unlikely to significantly quell unrest within the party.
A YouGov poll for the Sunday Times has found that nearly half (48 per cent) of Britons believed Ms May should resign.
The deal with the DUP is on a "confidence and supply" basis, which means the party would lend its support to block no confidence votes and pass budgets.
The DUP supports Brexit - but also wants to avoid a hard border with the Republic of Ireland and to maintain as far as possible the current access to European Union markets - both of which would be jeopardised if the United Kingdom leaves without a deal in place, an outcome known as a "hard Brexit".
"It would be nothing short of scandalous if the Tory party jeopardised the return of government in Northern Ireland and the Good Friday process in order to cling on to power at Westminster".
Without a majority, she could be forced to seek consensus on the approach she takes, potentially by performing a U-turn on single market membership and protecting the economy at the expense of new immigration controls.
The pound on Friday fell 1.7% against the USA dollar and 1.4 % against the euro.
This is more than the total of Labour, Lib Dems, SNP, Plaid Cymru and Green Party.
However, Defense Secretary Michael Fallon said on the Andrew Marr show he believed the Conservative Party and the DUP will be able to hold together in government. May told German Chancellor Angela Merkel on Saturday she plans to start Brexit talks, as planned, in a couple of weeks.
Davidson, who is gay and set to marry an Irish catholic later this year, subsequently said she had received "categoric assurances" from Downing Street that any deal would not affect LGBTI rights. This will become particularly hard to do once Brexit legislation comes before Parliament.
In the Sunday Mirror, opposition Labour leader Jeremy Corbyn, who scored hefty gains in the election, said there was still a chance for him to be prime minister if May failed to form a government.
Another poll by Survation, published in the Mail on Sunday, recorded 49 per cent of the population as saying she should quit, with 38 per cent saying she should remain in the post.
"Theresa May is a dead woman walking".
May had repeatedly ruled out the need for a new election before changing her mind.
The alliance makes some modernizing Conservatives uneasy.
He said he is "ready any time" for another vote and claimed he will lay down his own programme of policies after failing to give up on supplanting the Prime Minister. Ruth Davidson, the leader of the Scottish Conservatives, expressed those doubts publicly on Friday evening after speaking to May.
The statement also said that both leaders "agreed that of immediate concern were efforts to establish an executive as soon as possible, with exploratory discussions with the NI parties to take place tomorrow".