She suggested MPs could be "given a role" in deciding whether to activate the backstop, which is created to stop the return of a physical border.
Asked if Mrs May still felt she could command a majority in the Commons for the crunch vote next Tuesday, a Downing Street source said: "Everybody knows the parliamentary arithmetic".
Mr Graham said: "What we are trying to achieve is something that gets a lot of support from colleagues and that the government, we hope, will take forward because it will make a real difference to the vote".
The backstop is an arrangement created to ensure there is no return to a "hard border" between Ireland, which will remain part of the European Union after Brexit, and Northern Ireland when the United Kingdom leaves the bloc next March.
Led by Hugo Swire, the alternative plan would give MPs the chance to vote to extend the transition period at the end of 2022 - rather than falling into the backstop.
He said it meant the United Kingdom can continue to work with Brussels on cross-border investigations on modern slavery, using DNA databases to catch criminals, the fast-track extradition of suspects, along with working alongside Europol and Eurojust.
"There are questions about how decisions are taken as to whether we go into the backstop, because that isn't an automatic", May said.
May's critics, including both supporters and opponents of Brexit, say that means Britain could be subject to European Union laws long after it has given up any influence over determining them. Even if we get a deal next week the issue is far from over - it is just Act One.
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Ms Meadows added: 'Just over 19% of all the British citizens living in France are over 65, so a significant number of people are potentially at risk if their pensions should cease to be uprated in the case of no deal.
Those loyal to the prime minister have urged her to delay the vote, arguing that, at the moment, it is unwinnable.
Mr Cox's advice stated that, if Brussels felt that negotiations on a trade deal had broken down or were taking too long, it would be able to apply to the arbitration panel for Britain to be removed from the customs union while Northern Ireland remains - effectively creating the border in the Irish Sea, which Mrs May has said no prime minister could accept.
"The agreement that is on the table - the withdrawal agreement and the agreement on the future relationship - are, in our view, the only and best possible to organize an orderly withdrawal".
"We do this by removing each option in turn, and then looking at the second choice of people who backed that option".
However, it has already been rejected by Arlene Foster, leader of the Democratic Unionist Party, which is propping up Mrs May's government.
Chief whip Julian Smith acknowledged he faces an "uphill challenge" to persuade MPs to back Mrs May's deal, but insisted "it's all to play for". All this advice reveals is the central weaknesses in the Government's deal.
"The people made their decision and I back them by supporting a deal which delivers on the outcome of the referendum". However, the Labour Party is keen to push for a general election, something that could prove an uphill task given the UK's Fixed Term Parliaments Act setting out a five-year period between elections.