May will promise lawmakers a second opportunity to influence the Brexit talks later in the month in a bid to stave off any rebellion from within her own party by those who fear Britain could end up leaving without a deal.
The British parliament is set to hold a debate on Brexit on February 14 but this is not a re-run of a vote last month on whether to approve the exit deal Prime Minister Theresa May's negotiated with the European Union.
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 a member of May's cabinet pledged Sunday to give parliament a further ballot two weeks later - a measure meant to give the premier more time for talks with the EU.
"I am not clear why you believe it would be preferable to seek a say in future European Union trade deals rather than the ability to strike our own deal?"
Earlier today, Treasury Chief Secretary Liz Truss refused to rule out quitting if May shifted to a position of backing a customs union in order to get a deal through parliament.
"We can't allow that to happen", Sir Keir said.
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"You've got to decide which of those Brexits you want before we leave - otherwise, we're going to leave without clarity".
May is seeking changes to her deal with Brussels after it was rejected by a record majority in parliament on January 15.
Last month, Parliament voted in favour of an amendment that supported most of the PM's deal but called for backstop - which is a last-resort option to prevent a hard border in Ireland - to be replaced with "alternative arrangements".
"The government will commit that if the meaningful vote, in other words the deal coming back, has not happened by the 27th of February, then we would allow a further motion - votable in Parliament - to take place, to give that sense of assurance as to the process moving forward", he told the BBC's "Andrew Marr Show".
The wording of May's motion on Thursday, asking Parliament to back her negotiation, will be closely read. The prime minister is now in talks with Brussels to seek these changes to the backstop.
He told the Sophy Ridge on Sunday programme that it would lead to a "very hard border" on the island of Ireland and is contrary to the Good Friday Agreement, the peace deal he helped negotiate which brought an end to The Troubles.
Critics of the backstop in Mrs May's current deal say they could tie the United Kingdom to European Union rules indefinitely or mean Northern Ireland ends up under a different system to the rest of the UK.