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 Conservative backbencher revealed that six undercover police officers gave protection to an MP on a public engagement amid claims that threats were influencing Brexit votes.
Theresa May is facing one of the biggest tests of her leadership, with Tory rebels threatening to vote against their own party to force through amendments to the bill. The "meaningful vote" amendment, which would assert Parliament's authority over the negotiations and effectively hand control to MPs, still represents an opportunity to turn the whole process on its head. Similarly, the House of Commons also defeated an attempt by some government and opposition MPs to remove the "exit date" from the withdrawal bill.More news: Navarro Voices Regret for Harsh Words About Canadian Leader
The disagreement centres on whether the government agreed to consider a specific clause of the rebel proposal that would hand parliament control of the Brexit process if ministers are unable to strike an exit deal by February 15, 2019.
They said: "We have not, and will not, agree to the House of Commons binding the government's hands in the negotiations".
Asked on BBC Radio 4's Today programme what would happen under Government plans if MPs voted against the deal eventually secured by Mrs May, Mr Davis said: "If they throw it out, well, they throw it out".
That both sides have dug in so doggedly is a sign of how important - and divisive - the amendment is: it will determine whether parliament will have more power over the final stages of the Brexit deal, rather than the "take it or leave it vote" Mrs May has offered. Wollaston tweeted: "Following further assurances that further govt amendments will come forward in the Lords, I will now be supporting the govt".More news: Wonder Woman 2: Patty Jenkins teases return of key character
Brexiteers were dejected by the turn of events, but are pinning their immediate hopes on the detail in the government's compromise.
One government official said: "It's not over yet".
Keir Starmer, Labour's shadow Brexit secretary, was refusing to accept the government at its word Tuesday. "We will wait and see the details of this concession and will hold ministers to account to ensure it lives up to the promises they have made to parliament".More news: United States air force deserter found alive in California after 35 year manhunt