Theresa May's government will collapse if MPs vote against her Brexit deal, a senior Tory has said.
After the vote, Mr Grieve told Sky News: "I've voted with the Government following a meeting with that our concerns will be addressed".
Asked about what had been promised, Mr Buckland, the solicitor general, said the government remained "open-minded" but he would not "blithely" commit to any changes until he had had those conversations.
On Wednesday, May had faced the prospect of losing a vote after rebels had indicated their support for a change introduced by the House of Lords to require ministers to report what efforts they had made to secure a customs union.
In a day of drama, May's position seemed suddenly weaker when junior justice minister Phillip Lee, who has always been critical of Brexit strategy, resigned and said he would vote against the government.
The House of Commons is debating a crucial bill to pave the way for Britain to exit the European Union next March.
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.
Laura Smith, MP for Crewe and Nantwich, was among 15 Labour MPs who voted with Theresa May to reject the EEA amendment.
The first part of Grieve's plan means that if parliament votes against the government's deal, ministers have 21 days to come back with a plan of action, which must then be approved by a vote in parliament within seven days.
"Whatever we do, we're not going to reverse that [decision to leave the EU]", he said.
But she faces a gruelling bout of "parliamentary ping-pong" with the Lords, as the Bill bounces back and forth between the two Houses over the coming weeks.
PARLIAMENT TV DIVIDED Corbyn told his MPs to abstain on the House of Lords amendment
He joined four other Labour MPs who quit the party's frontbench to back an amendment from the House of Lords that would keep the United Kingdom in the EEA.
Labour's Brexit policy chief, Keir Starmer, said May had been forced to avoid a "humiliating defeat" and "to enter negotiations with her backbenchers".
She added: "I can not countenance parliament being able to overturn the will of the British people".
On Tuesday, May averted a potentially devastating defeat by convincing the group - who were prepared to vote her down - to discuss new wording of what it would take for Parliament to grab control of the Brexit process. View of Lab MPs is clear, we don't want a hard Brexit'.
May objected to the amendment - inserted by the House of Lords - because she said it would tie her hands in the negotiation.
May's government is divided between Brexit-backing ministers such as Foreign Secretary Boris Johnson who support a clean break with the European Union, and those such as Treasury chief Philip Hammond who want to keep closely aligned to the bloc, Britain's biggest trading partner.
"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".
"Time will tell as to whether this is just another attempt to buy off the rebels or a real attempt at consensus".
Ukip leader Gerard Batten said: "The only "meaningful vote" was the verdict of the people in referendum of 23rd June 2016".
Senior Tory Nigel Evans added: 'Bracknell folk voted out by a greater margin than the United Kingdom - we all stood on a manifesto only a year ago to deliver brexit and that is what the people now expect.
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