But these putative pledges by the PM are inconsistent with Tuesday night's statement by Davis's officials that any new amendment relating to the power of MPs to accept or reject a Brexit deal must not restrict her negotiating freedom - or restrict her ability to sign whatever treaty with the European Union she would like.
The Government has won the first votes during today's Brexit showdown.
They subsequently warned they will not be easily bought off, while Tory Brexiteers urged ministers not to concede any ground.
Remain-supporting Conservative MPs had threatened to back an amendment to the bill which would have given parliament a more widespread veto. Mr Grieve's proposal will now be added to the legislation.
Dr. Sarah Wollaston, chair of the heath select committee, switched to back the Government after saying she was "minded" to rebel over voting against the Lord's amendment.
"We have not, and will not, agree to the House of Commons binding the government's hands in the negotiations". "It was the prime minister who I sat in front of this afternoon and who gave us those assurances".
Her fellow Conservative backbencher Stephen Hammond said: "Parliament must be able to have its say in a "no deal" situation".
"In all conscience, I can not support the Government's decision to oppose this amendment because doing so breaches such fundamental principles of human rights and Parliamentary sovereignty".
To buy off a group of Tory rebels - whose ranks were boosted by the shock resignation of Justice Minister Phillip Lee Tuesday morning - the prime minister agreed in principle, according to the rebels, to write into law a new deadline in the Brexit talks: November 30, 2018.
"This justifies my decision to resign and makes it a lot less painful".
Among the 14 amendments to the Bill - set to be voted on by MPs on Tuesday and Wednesday - are changes which would see the United Kingdom stay in the Single Market and would allow Parliament to dictate future negotiating terms.
Well what the Remainer MPs thought they heard from May does not seem compatible with Davis's red lines.
Passions ran high in Tuesday's three-hour debate, when angry eurosceptics accused their rivals of trying to undermine the 2016 referendum vote to leave the EU.
Brexit Secretary David Davis has warned Conservative Party rebels that proposals to give Parliament the power to direct negotiations with the European Union are simply a tactic to overturn the results of the 2016 referendum that mandated Britain's departure from the bloc.
"That term, Mr Speaker, is actually specifically defined in the Belfast Agreement and I've no doubt that government ministers will have read the Belfast Agreement in its entirety", she said.
The final vote, as outlined in Grieve third point, would be different as the government would then have to follow any direction given by the Commons.
It sets up the prospect of frantic efforts by Mrs May to appease both Leave and Remain sides of her parties, as well as another potential crunch vote in the House of Commons next week.
The Bracknell MP, who called for a second referendum on whatever deal Mrs May secures from the European Union, later told the Commons there was growing evidence that the Government's Brexit policy is "detrimental to the people we were elected to serve".