Buckland indicated the government would discuss the possibility of adopting Grieve's push for ministers to secure parliamentary approval for their Brexit plans if they fail to negotiate a deal with the EU.
Senior Remainer Dominic Grieve said Mrs May promised to table amendments in the House of Lords which will address their concerns.
"I expect the government to honour its commitments and I expect the PM to honour her commitments and I have no reason to distrust the approach she took with us", Mr Grieve told the BBC's Newsnight.
'I hope very much the government will look at that, because I think it provides a solution which would satisfy everybody, ' he said.
After days of division and bitter rows between Brexiteers and Remainers in the Tory ranks, Mrs May was able to see off another amendment that would have tied Britain to the customs union post-Brexit by 325 votes to 298, majority 27.
But while that vote seemed assured, tensions over Britain's departure from the European Union boiled over in parliament, where lawmakers from the Scottish National Party walked out in the middle of questions to the prime minister in protest at what their leader said was Scotland being ignored in the Brexit debate.
Theresa May's landmark Brexit legislation returns to the House of Commons on Tuesday, with the 15 amendments added to it by the upper House of Lords mostly created to keep closer ties to the European Union and give Parliament more power over the divorce process.
"MPs are now voting to remove another exit day amendment". It would reduce the risk of the United Kingdom exiting the European Union without a deal, as it means that MPs could insist that the government go back to negotiating table.
"Whatever we do, we're not going to reverse that [decision to leave the EU]", he said.
But part C - arguably the most controversial section of the proposal - has been pushed back, with Theresa May apparently promising to have further discussions in the Lords and Commons as ping pong between the two Houses of Parliament continues.
In the event, Dr Lee abstained on the crucial vote, saying he was "delighted" that the Government had agreed to introduce an amendment giving Parliament "the voice I always wanted it to have in the Brexit process". "I am sure a sensible amendment will be forthcoming which we can all agree to".
But Brexit Secretary David Davis' department was defiant.
Davis says the government can not accept amendments to Brexit bill that allow Parliament to instruct it on steps to take in an global negotiation.
While promising "further discussions", he said he was concerned that empowering Parliament to "instruct" ministers what to do in the event of no deal would leave the United Kingdom in "very rocky constitutional territory".
Liberal Democrat Brexit spokesman Tom Brake said: "At the 59th minute of the 11th hour, as has become a tradition in Brexit negotiations, the Tories have been forced to cobble together a compromise".
Ms Allen insisted that the referendum was a binary vote and it didn't say that MPs should neglect their duty, but Nick hit back: "Your duty, some would argue, is to deliver Brexit, which is what the people have told you to do".
But the concessions were the latest manoeuvre by a minority government that has been forced to compromise with parliament or to simply put votes off until a later date because of its inability to force through the policies it backs.