Labour will seek to force Mrs May into a decisive second Commons showdown on her Brexit deal by February 26.
Shadow Brexit Secretary Sir Keir Starmer [PA Wire] However, MPs are now set to be promised another vote if a new deal hasn't be struck by 27 February, where they will also be able to make suggestions - including extending the Article 50 deadline.
When asked if a meaningful vote would happen this month, he said: "If it has not happened, parliament would have further opportunity by no later than 27 February".
As the clock ticks down to Britain's scheduled exit on March 29, May is trying to persuade the European Union to change a deal that was agreed between London and Brussels late a year ago but overwhelmingly rejected by parliament in January.
Thirdly, and most pertinently, he said the European Union is willing to listen to Mrs May's ideas on how to end the impasse and expressed his hope she would be able to put forward a realistic suggestion and provide "a believable guarantee for peace in Northern Ireland" that prevents the return of a hard border.
He acknowledged that more work was needed to get the United Kingdom ready for Brexit on March 29, telling the BBC's Andrew Marr Show there are "still steps that are currently being put in place" but "there is steady work that is going on, 10,000 civil servants that are now focused on this" and the Border Force was "ramping up" its staff. "There needs to be a day when Parliament says that's it, enough is enough".
"We have got to put a hard stop into this running down the clock".
Labour leader Jeremy Corbyn has written to the Prime Minister setting out five demands that would have to be met for his MPs to support a deal, including a permanent customs union and close alignment with the single market.
The move led to a backlash from pro-EU Labour MPs, but Mr Starmer defended the approach and warned against a split in the party.
"It seems to me we're now at the point where we can have meaningful talks to get a deal between the main political party leaders", he said.
Housing minister James Brokenshire said on Sunday May would commit to giving parliament another debate on Brexit with the chance to vote on alternative options, if a deal had not yet been agreed and voted upon by then.