Senior Tories have lined up to say Theresa May will not face a leadership contest but a former minister warned she must not lead them into another election.
The Prime Minister's authority has been severely damaged by her snap election gamble, which resulted in the party losing its Commons majority and reliant on the votes of the Democratic Unionist Party.
With the imminent start of Brexit talks and no appetite for another general election, it appears Mrs May is safe from an immediate challenge.
Former minister Lord Barker said the mood of the party when MPs arrive in Westminster on Tuesday is set to decide her fate.
But he told BBC's Newsnight: "I detect that there is no appetite in the party for an immediate leadership contest.
"So we will have to see what the Commons has to say when they meet."
While Mrs May is set to be allowed to govern, she will have to go by the time of the next election, he said.
"There's clearly not going to be another election with Theresa May at the head.
"We have discovered that she is a competent minister, potentially a tough negotiator, but a terrible campaigner," he said.
Lord Barker tipped Home Secretary Amber Rudd as a potential successor.
Ex-cabinet minister Owen Paterson warned his party against the "chaos" of a leadership contest as it could further delay resolving Brexit matters.
But the MP was less sure about how Mrs May could continue to rule with the DUP, saying "let's see how it pans out" when asked.
Speaking on BBC Radio 4's Today programme, the former Northern Ireland secretary said: "We are nine days off from the Brexit talks starting, we've had two horrific security incidents, and to cast the Tory Party into yet another leadership contest I think would be very unwise."
Former Tory leader Lord Hague said "very serious lessons" had to be learned by the party but warned against a leadership contest.
Writing in The Daily Telegraph, he said: "Voters do not want further months of uncertainty and upheaval.
"They want to see ministers getting on with the job, while acknowledging democracy and their constrained circumstances."
Former Tory health minister Nicola Blackwood, defeated by the Liberal Democrats in Oxford West and Abingdon, said "no party has been a winner" at the election.
On Mrs May's future, Ms Blackwood told Today: "She has to have the right team around her but they need to support her and make sure they put aside partisanship right now because it's about making sure we survive the security challenges we face but also the Brexit negotiations."
Former minister Gavin Barwell, who lost his Croydon Central seat to Labour and has now been appointed as Mrs May's chief of staff, said: "I don't think you can call a result where you have got a vote share up at nearly 43% disastrous."
Reminded that the party had lost seats, Mr Barwell replied: "I'm not trying to pretend it's a triumph, clearly we didn't get the result we hoped to get.
"I'm just putting on record that she did better than (John) Major."
Asked if Mrs May has the authority to run the party and country, Mr Barwell said: "I believe there's a will in the Conservative Party to get behind her and support her."
A majority (59.48%) of Tory members now believe Mrs May should resign, according to a snap survey by the ConservativeHome website, while 36.66% of the 1,503 respondents believe she should stay on.
Former minister Ed Vaizey indicated Tory MPs were actively discussing Mrs May's position using the WhatsApp messaging system.
Asked if there were phone calls being made between Tories about the next leader, Mr Vaizey told the BBC: "That's so 20th century. It's all on WhatsApp."
He added: "We all talk on WhatsApp ... lots of MPs are in lots of different groups."
Mr Vaizey said he hoped the election result would soften Mrs May's stance on Brexit, adding: "I hope in the next few days we will see a clear acknowledgement that a 'no deal is better than a bad deal' is off the table, that we are going for a Brexit that is going to secure jobs and investment."
Tory MP Philip Davies joked: "As I have been told for years that I am the Tory equivalent of Jeremy Corbyn then perhaps I should now consider a leadership bid."
Some Tories expressed concerns about the prospect of closer links with the DUP, which opposes gay marriage and abortion.
Totnes MP Sarah Wollaston insisted any deal with the DUP must not be allowed to influence policies on the death penalty, gay rights, education or abortion.
In a series of tweets she said: "I will always oppose the death penalty & would resign if others imposed it. They won't.
"I will always support the right for women to choose & access safe termination of pregnancy & will oppose any change to the legislation.
"I will never agree to any dilution of LGBT rights.
"Creationism in schools? Hell no.
"If any of that is a condition of confidence and supply it simply won't work."
Heidi Allen, MP for South Cambridgeshire, said she was "deeply unhappy" with the idea of a formal coalition with the DUP.