"She said she will serve us as long as we want her."Lawmakers, who are by tradition not named at such meetings, told Reuters that there were no dissenting voices and that the party had no appetite for a leadership election".
Ms May appeared contrite, sought to apologise for her failed election gamble and gave an explanation of what went wrong.
"She said 'I'm the person who got us into this mess and I'm the one who is going to get us out of it, '" said one Conservative lawmaker who attended.
May moved to demonstrate that she understands the frustration of voters by moving up a meeting with rank-and-file Conservative Party lawmakers, some of whom have called for her to step aside sooner rather than later.
Her new chief of staff, former Tory MP Gavin Barwell received large cheers from MPs, who had been unhappy with the former team around the prime minister.
May's party fell eight seats short of retaining their parliamentary majority, and is now in talks with Northern Ireland's ultra-conservative Democratic Union Party (DUP), which won 10 seats, to forge an informal alliance. Despite anger at the election, she was cheered briefly at the start of the meeting.
The meeting came shortly after First Secretary of State Mr Green said that agreement with the DUP would have to be sealed before finalising the details of the Queen's Speech, setting out the Government's legislative programme.
Mrs May signalled that she still meant to serve a full term. She has called for a closer relationship with the European Union after Brexit."There can be changes in the offer of Brexit as we go forward", Davidson told reporters in London after meeting May.
"The reality of this Parliament inevitably will be one of a great deal of consultation, a lot of work trying to build alliances, probably even on individual aspects of legislation, to try to pull in support from people in other parties".
Boris Johnson, the Foreign secretary, is also said to be considering a leadership bid, although he has played down suggestions and is said to want to bide his time until Mrs May decides to leave of her own accord.
"The key issue is to ensure we put our country's economic future first and foremost in our minds as we go ahead with Brexit".
Confusion descended into farce after it was suggested the speech could not take place for up to a week after it is finalised because convention requires it to be written in ink on goat skin parchment, which takes a considerable amount of time to dry.
A Labour spokesman said that uncertainty over the date of the State Opening showed the Government was "in chaos", while Liberal Democrat leader Tim Farron said it was "an utter humiliation" for the PM.
It could also allay fears of disruption to complex supply chains for manufacturers, but a customs deal on goods would do little to help London's big banking and insurance industries, which face exclusion from the EU-regulated market in services. However, the MPs said they felt more able to ignore the concerns when they realised many emails were due to a co-ordinated lobbying campaign by the left-leaning campaign group 38 Degrees.
But Mr Green said there was "complete unanimity" in the Cabinet and "overwhelming support" for Mrs May's approach.
DUP leader Arlene Foster said there had been "positive engagement" so far.
The Prime Minister was forced to perform an unprecedented U-turn within days of the publication of the Tory manifesto by announcing there would be a cap on social care costs, something that had been absent in the original policy document.
"We are all agreed that that would be a very bad idea and the prime minister agreed", said a former cabinet minister, who has not been overly supportive of the prime minister, following the make-or-break meeting of backbench Tories on Monday afternoon.
May also promised her MPs that there would be a "more open door policy" towards backbenchers.
"Theresa May can squat in Number 10 as long as she wants but the message is stark - she has no power, no influence and her game is up", said Mr Farron.