It is not a foregone conclusion that lawmakers will back the amendment, one of seven put forward to try to break the impasse in Parliament.
May told Parliament after the vote that "it is now clear that there is a route that can secure a substantial and sustainable majority in the House for leaving the European Union with a deal".
"We will now take this mandate forward and seek to obtain legally binding changes to the Withdrawal Agreement that deal with concerns on the backstop while guaranteeing no return to a hard border between Northern Ireland and Ireland".
He said there were more downside risks for sterling in the days ahead.
"The Prime Minister has been aware of the discussions".
This proposal, tabled by Labour's Yvette Cooper and Tory Nick Boles, is one of the strongest-backed of a bevy of rival amendments tabled for debate. Sir Graham is the chairman of the influential 1922 Backbench Committee, the body of Conservative members of Parliament who don't have a job in the government.
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A Winter Weather Advisory and a Winter Storm Warning are in effect across most of the region through Monday at 6:00 p.m. Temperatures at night are expected to be about 25 below with wind chills of 48 below in McHenry County, she said.
May already tried and failed to wrest new concessions from Brussels during a hard summit last month that exposed her weak negotiating position - and the European Union's resolve.
The EU's deputy chief negotiator, Sabine Weyand, said on Monday that the bloc was "open to alternative arrangements" on the Irish border, but Brady's proposal did not spell out what they were.
They have said that they will only back May and the Brady Amendment if she is willing to commit to work together with them on their plan, which is to replace the current backstop.
Pound reaction: Anything that means the European Union has to re-open negotiations would be taken negatively, according to Russell Silberston, a portfolio manager at Investec Asset Management. But traders said that fears of a no-deal Brexit were driving the pound down. "But in contrast to a fortnight ago, this house has made it clear what it needs to approve the withdrawal agreement".
Amendments that could alter the path of Brexit altogether are what really matter.
But some MPs from the PM's own backbenches, who backed Remain in the referendum, will support rival proposals to try and rule out the United Kingdom leaving the European Union without a deal. "The problem with the Brady amendment is that it does not spell out what they are".
May will be wishing there are more like him when, as is likely, she comes back from Brussels empty handed.
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This time, he was in a tie for fifth place with Northern Ireland's Rory McIlory (69) and first-round leader Jon Rahm (72) of Spain.
If Labour whips its support, May's minority administration will need most of its own rebels to fall into line while hoping that Labour MPs who are concerned at being accused by voters of frustrating Brexit will abstain or vote with the government.
Trickett said he felt "it may look to people as if we're trying to somehow remove the earlier decision, which was to Brexit", by voting for an amendment meant to extend article 50. It is this and other plots like it that high profile Brexiteer Boris Johnson has accused those trying to delay Brexit as actually drawing a "thin disguise for killing it off".
Amendment G: Proposed by Conservative MP Dominic Grieve.
Downing Street is desperate to prevent Cooper's amendment passing.
Another defeat for pro-Europe politicians, this time Ian Blackford's Scottish nationalist amendment.
It "rejects the United Kingdom leaving the European Union without a Withdrawal Agreement and a Framework for the Future Relationship". "(They) have the vote tonight and then we will take it from there".
DUP leader Arlene Foster gave her party's support to the Malthouse Compromise.
Yellow weather warning issued for snow and ice
Highest temperatures of 3 to 7 degrees with mostly light west to northwest breezes. Sleet and snow showers are likely, particularly in northern and eastern counties.