However, pro-EU Tory MPs have warned they are ready to vote down the plans for what should happen in the case of a no-deal Brexit. The question is by how much?
"We won't be accepting the Lords amendment", the source said, referring to a decision in the upper house of parliament on Monday to again try to force the government to hand the House of Commons more control over Britain's exit from the EU.
Grieve added: "One of the reasons I've supported [the amendment] is precisely to avoid the sort of crisis a government would immediately collapse". He said the compromise reached was changed and made it unacceptable.
"Ministers, the prime minister in particular, have promised a meaningful vote, that promise has not been honoured", former minister Douglas Hogg, who said he had been in talks with Grieve since last week, told lawmakers as he introduced the amendment.
Crucially, the motion will be unamendable, meaning MPs can not insert a requirement for Mrs May to go back to the negotiating table, extend the Brexit transition or revoke the UK's withdrawal under Article 50.
"If the Prime Minister's final Brexit deal is voted down, that can not give her licence to crash the United Kingdom out of the European Union without an agreement", Labour's Brexit spokesman Matthew Pennycook said in a statement.
After an ill-judged election past year, May relies on the help of a deal with a small Northern Irish party to win votes in the Commons and can afford to lose no more than a handful of Conservative rebels if she is to avoid an embarrassing defeat.
But she added: "Parliament can not tie the hands of government in negotiations".
The peer said he was asking the Lords to allow MPs to vote on what Mr Grieve believed was agreed with the Government.
Anti-Brexit demonstrators wave EU and Union flags opposite the Houses of Parliament, in London, Britain, June 19, 2018.
The pro-EU Conservative lawmakers want parliament to be able to intervene before Britain's deadline to leave the EU by next March, if May's government either fails to negotiate a deal with Brussels, or if it reaches a deal that lawmakers reject.
The government then has two weeks to return to the Commons with a statement on how the government plans to proceed.
May has said that in the week beginning July 9 the government would set out in more detail than ever before its ambition for future relations with the EU. "MPs now face a decisive vote to guarantee Parliament has a proper role in the Brexit negotiations and to take the threat of no deal off the table once and for all".