The pound rose near $1.293 at around 4 p.m. London time after trading at $1.286 for much of the session. However, it quickly fell back to around 1.287 moments afterwards.
The political editor of ITV News, Robert Peston, tweeted that an influential group of pro-Brexit Conservative politicians could back the deal at a parliamentary vote on Tuesday evening. This would only happen, he said citing an anonymous MP, if an opposition Labour lawmaker dropped an attempt to block the U.K. leaving without a formal deal.
May's proposal is split into a "Withdrawal Agreement," setting out the terms of the divorce and a "Future Relationship" document, which drafts how the U.K. will interact with the EU in the future. The U.K. government and EU heads of state have signed off on the preliminary deal and now U.K. lawmakers have a "meaningful vote" to decide whether it can proceed.
Sterling's rise on Monday was quickly halted by a denial from a member of this euroskeptic group of Conservatives, known as the European Research Group (ERG).
"We plan to vote no to everything: all amendments and the main motion, whether or not amended," Conservative lawmaker Steve Baker said on Twitter.
—CNBC's Sam Meredith contributed to this report.