Labour MP Jim Fitzpatrick has said he and his colleagues will back a deal brought before the Commons in order to "get Brexit done and dusted".
Mr Fitzpatrick is one of 19 Labour MPs who have written to the European Commission president Jean-Claude Juncker urging him - and the government - to show more flexibility in negotiations.
He says the reason for sending the letter was to "put pressure on both sides" to come to an agreement, after talks between the UK and the bloc reportedly began to collapse yesterday.
"There are a number of Labour MPs in the Commons who, like most people, want to get this finished," he told talkRADIO's Julia Hartley-Brewer.
"If the EU and Number 10 can arrive at an agreement, there are enough people who would be prepared to vote for this and put it to bed."
He continued: "If there is an agreement, there are Labour MPs who in the national interest will vote to support a deal to get Brexit done, and then get on to the serious negotiations about all the different arrangements which have to be sorted out over the next few years."
The Poplar and Limehouse MP said there were "certain arrangements" in Mrs May's Brexit deal that were "acceptable", but branded Boris Johnson's current Brexit proposals as "unacceptable".
"There are people within the European Commission and among the EU 27 who do not think there is a majority in the House of Commons for any deal whatsoever, so they're playing a long, good game. They're trying to string us out. They're trying to support another extension," he added.
"There are some people, no doubt, who think they can block Brexit completely and it's going to go away."