A similar motion will be tabled on Thursday if, as seems inevitable, May has not yet agreed enough changes to her overwhelmingly rejected departure plan to put it to another vote. The new pledge would be for a repeat of this process by February 27.
Labour's Brexit policy chief Keir Starmer told the Sunday Times newspaper that his party would seek to use the debate in parliament this week to prevent May from waiting until the last minute to come back with a deal, and compel her to present a fresh accord for lawmakers to consider before February 26.
Britain is due to leave the European Union on March 29th but Parliament has rejected May's divorce deal, leaving the Prime Minister to seek changes from a resistant EU.
But there is no commitment to hold a binding vote on the deal itself by the end of the month.
Communities Secretary James Brokenshire said Sunday that parliament would get another chance to pass judgment on May's Brexit plan "by no later than February 27".
After talks with Brussels this week Mrs May said that she was determined to deliver Brexit on time ahead of the March 29 deadline.
May is already due to update parliament on her progress toward a deal on Wednesday and then on Thursday to give parliament a chance to express their opinion.
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.
'We can't allow that to happen, ' Sir Keir said.
It comes after the Labour leader Jeremy Corbyn, who has written to the Prime Minister setting out his demands for a Brexit deal he could support, accused Mrs May of an "utterly cynical" approach. "That isn't right in terms of the respect for parliament", said Starmer.
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.
In his letter to Mrs May, the Labour leader set out five demands, including a permanent customs union and close alignment with the single market.
"It will be the decisions that businesses take about jobs and investment - and they will reduce them - so you have less potential to trade globally, that will mean less investment in the future and that will mean fewer jobs in the future", she told Sky News.
May and her government have repeatedly said membership of a customs union would prevent it having an independent trade policy - something they have promoted as one of the main economic benefits of leaving the EU.