The move by Sir Graham Brady, who chairs the 1922 committee, is a significant shift and came as senior government sources predicted that May could hold the crucial meaningful vote as early as this week.
"The possibility that monetary tightening might be needed in the future does not necessarily mean we need to tighten now", Saunders said in a speech at Imperial College in London.
The government will announce plans to write a promise of improved consultation on post-Brexit workers' rights into legislation and offer lawmakers, businesses and trade unions a say on whether Britain should match any future EU rights laws.
May has ruled out any such union which would be an anathema to eurosceptic MPs in her Conservative Party, although the idea has the support of some pro-EU Conservatives and the Labour Party.
Fox said it would be "very unfortunate" if MPs were to reject May's deal and then vote to extend Article 50, in votes the prime minister has promised will take place by March 14.
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The group wants a legally binding, treaty level arrangement, the language must not just simply reiterate the temporary nature of the backstop, and there "needs to be a clear and unconditional route out" of the arrangement. "But from day one we have been very straight about a few things and among those is that Brexit can not lead, under any circumstances, to the emergence of a hard border between Ireland and Northern Ireland".
"I have always said whatever happens in negotiations, we are a strong country and will find a way to prosper".
Getting the backing of the ERG for her deal would be a major boost for Mrs May but would still not guarantee she could get it through Parliament. Should that fail, as expected, there would be a vote a day later on a "no-deal Brexit". And there are a variety of views within the ERG itself, with some leading figures taking a more hardline approach than others.
But pro-Brexit Labour lawmaker John Mann, who has discussed the funding plan with May, said it meant 'new money, which will need a lot more from existing sources of funds and the replacement for European Union structural funds.
Labour has so far failed to get MPs to back its alternative Brexit plan, which would see the United Kingdom in a permanent customs union with the EU, and has now said it will attempt to get a new EU referendum.
Labour's shadow chancellor, John McDonnell, was also quoted as saying that few Labour MPs would vote for May's deal.