Committee chairman Graham Brady said she was right to take "some responsibility" after calling the snap election and leading "one of the worst campaigns I can recall".
There are two ways a contest can be triggered, most obviously if the leader of the party resigns.
Another Tory backbencher said the reception for the PM was "good".
"The people of Britain have had a bellyful of promises and politicking". Another minister told me in the medium term it is "hard to see how" the PM can stay.
Details of which elements of the manifesto might be dropped have not been confirmed, but it is thought that Mrs May's poorly-received proposals for funding social care and her cherished ambition to open new grammar schools may have to be ditched.
However many Tories are adamant that she can not lead them into another general election after her disastrous showing last week at the ballot box.
Who are the winners and losers in Theresa May's cabinet reshuffle?
Mr Barwell has replaced Mrs May's key aides Nick Timothy and Fiona Hill, whose resignations were reportedly demanded by Tory MPs as the price of their supporting the PM.
It had been her decision to call the election, and her leadership of the campaign that faltered and in what was described as the most human performance they had seen for quite some time, Theresa May said sorry, time and again.
Defence Secretary Sir Michael Fallon disclosed that senior ministers had confronted Mrs May and told her she had to change the way she operates.
Mrs May's hopes of survival hinge on a deal with the Democratic Unionist Party to provide her with the necessary support she needs in the Commons.
Mr Davis told BBC Radio 4's Today programme: "We are being given an instruction by the British people and we've got to carry it out".
Meanwhile, Scottish Tory leader Ruth Davidson said Ms May should work with other parties to form a consensus on Brexit, having met the Prime Minister at Downing Street.
There was a "huge amount of work to get on with", said Mr Green, adding: "Not just the Brexit negotiations that start next week, but many other challenges that face us and we are determined to produce a Queen's Speech to ensure we can fulfil the Prime Minister's ambition to have a country that works for everyone".
Westminster was rife with speculation that the State Opening of Parliament may be delayed, after the Prime Minister's official spokesman declined to confirm it would go ahead on the scheduled date of June 19.