Meanwhile, EU chief Brexit negotiator Michel Barnier said he does not believe the United Kingdom will have enough time to approve Mrs May's withdrawal deal by the scheduled exit date of March 29.
The UK voted for the measures at European Union meetings but ministers say it will now be up to Parliament to decide whether to implement them.
She plans to hold a second vote on the deal coming week after promising to seek changes to a controversial "backstop" arrangement to guarantee and open Irish border after Brexit.
The two sides are at a "critical point" in these negotiations, a spokeswoman for May said on Tuesday.
With Britain due to leave the bloc on March 29, May is seeking assurances on the so-called backstop arrangement aimed at preventing a return to hard border controls between European Union member Ireland and British-ruled Northern Ireland.
"There are no documents for us to examine at this stage, but we look forward to seeing in due course what the Attorney has agreed, so that we may assess whether it meets the requirements of the Brady amendment, which commanded a majority in the House of Commons and calls for significant, legally-binding changes to the Withdrawal Agreement", Mr Tomlinson said following the first meeting of the group at Westminster on Monday.
Also raising hopes are the softening positions of several hardline Brexit supporters in May's Conservative party who have dropped their demand that changes to the backstop be made to the withdrawal treaty itself.
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He explains: "This is not a time to make the best the enemy of the good, and most MPs are in a mood to compromise, but the danger of this backstop becoming permanent is a real one and it has to be tackled".
The ERG has set out three tests by which they will judge Mr Cox's efforts to secure changes to the backstop.
Mr Brokenshire told BBC Radio 4's Today programme that the money was not linked to getting the Brexit deal through.
Hunt said avoiding an indefinite backstop was the crucial issue, when asked whether London was sticking to either a time limit or a unilateral exit mechanism from the backstop.
"I think she'd be lucky if she got 15 Labour MPs", Nandy added.
Mrs May's hopes of winning over Tory Eurosceptics hinge on Mr Cox being able to change his legal advice about the indefinite nature of the backstop, which is meant to ensure there is no hard border on the island of Ireland in the absence of a future free trade agreement.
The country is on course to leave without an agreement if British MPs fail to approve a divorce deal struck by British Prime Minister Theresa May that was overwhelmingly rejected in a first vote in January.
Sir Graham, who said the government's handling of the Brexit talks had been "lions led by donkeys", added there was nevertheless a growing desire in the Commons to get behind a deal.