Graham Brady, who chairs the influential 1922 committee of Conservative members, said there was no public appetite for a second election after May's party failed to win a majority in Thursday's vote.
Support from Northern Ireland's Democratic Unionist Party (DUP) would allow her Conservatives to reach the 326 seats needed for a majority in parliament.
"We welcome this commitment, which can provide the stability and certainty the whole country requires as we embark on Brexit and beyond".
According to the statement, the "details will be put forward for discussion and agreement at a Cabinet meeting on Monday". In 2016, they had won 331 seats out of a total of 650, but this time they could only manage 318 seats.
British Prime Minister Theresa May's two chiefs of staff have resigned in the wake of the Conservative Party's disastrous election result.
A spokesman for Mr Johnson said: "The Foreign Secretary is 100% supporting the PM and working with her to get the best deal for Britain". Although Arlene Foster said her party fundamentally rejected the endorsement of anyone involved in paramilitarism in response to the Loyalist Communities Council backing of the party.
'I tell you this, we will absolutely remain part of the European Convention on Human Rights and European Court of Human Rights - we are not walking away from those vital post-war agreements that were made'.
The party's calls for increases in the personal tax allowance, continued rises in the national living wage, renewal of the Trident nuclear deterrent and revisiting terrorism laws are all similar to Conservative policy.
Mrs May needs support in Parliament because the Queen's Speech setting out the Government's programme is due on June 19, with a crucial vote on it expected after a few days' debate. But the senior Conservatives members, although very much against May's sustained rule, are of the opinion that it might lead to the rise of the Labour party and hence May must not go, not yet. She's hitched the Conservative Party's fortunes to the politics of the DUP, and every day that she clings to power with this arrangement she tarnishes the Conservatives' brand.
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That said May had "spoken with the DUP to discuss finalising" a deal in the coming week.
May's gamble in calling an early election backfired spectacularly, as her Conservative Party lost its majority in Parliament.
British voters dealt May a devastating blow in the snap election, called to strengthen her hand in Brexit talks, wiping out her parliamentary majority and throwing the country into political turmoil.
"I can still be Prime Minister".
The Democratic Unionist Party (DUP) agreed to prop up Theresa May's government last night, striking an outline deal that will see the group's 10 MPs vote with the Tories on budget measures and other key issues.
"I don't think Theresa May and this government have any credibility", Corbyn said.
May had called the election to give her a mandate for her strategy to take Britain out of Europe's single market in order to end mass migration from the continent.
Sinn Féin's leader in Northern Ireland Michelle O'Neill said the DUP had "betrayed the interests of the people" there.
British Prime Minister Theresa May speaks watched by her husband Philip in 10 Downing street, London, as she addresses the press Friday, June 9, 2017 following an audience with Britain's Queen Elizabeth II at Buckingham Palace w.
Ruth Davidson, the Conservative leader in Scotland, said she had asked May for assurances that there would be no attack on gay rights after a deal with the DUP.