Prime Minister Theresa May will pledge this week to give parliament another chance to voice their opinions on Brexit by February 27 as she tries to buy more time to negotiate a new deal with the European Union.
Referring to her letter to Labour leader Jeremy Corbyn, Sir Keir said the key question is whether Mrs May is prepared to move her red lines on the Brexit deal.
The Labour MP said that Parliament will have to intervene with a "hard stop" over concerns Mrs May will attempt to leave MPs with a choice of accepting her deal or crashing out of the European Union without any agreement.
Theresa May has effectively ruled out Labour's ideas for a compromise Brexit plan, shutting off another potential route to a deal as business groups warned that with less than 50 days to go the departure process was entering the "emergency zone".
The prime minister struck a conciliatory tone in her response overnight and said she looked forward to the two parties meeting again "as soon as possible" to discuss ways forward on Brexit.
Mr Johnson accused Mr Corbyn of trying to "frustrate" Brexit by demanding a customs union, which he said would prevent the United Kingdom from striking its own trade deals.
Britain is due to leave the European Union on March 29, but lawmakers in Parliament have emphatically rejected Mrs May's divorce deal, raising the prospect of a no-deal exit from the bloc unless more parliamentary support is found.
It was one of five demands he said May must meet if she wanted to secure his backing for any revised Brexit deal.
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Labour MP Lisa Nandy told the BBC's Politics Live that there were between 40 and 60 of her colleagues "who are actively looking for ways to support" a revised Brexit deal.
"We shouldn't be put in a position where the clock is run down and the prime minister says it's either my deal or even worse".
Theresa May is due to update the Commons later on the latest developments in negotiations with Brussels and Dublin, as the sides try to find a way through the impasse on measures for the Irish border.
At the meeting, Department for Trade officials were probed on the pledge by Mr Fox that he would sign free trade agreements to replicate the EU's 40 existing deals with countries around the world.
Mr Johnson said Mrs May's priority should be to secure a time limit on the backstop - meant to prevent the return of a hard border on the island of Ireland - before the next election scheduled for June 2022.
"I think it must be pretty obvious that if you are going to have a time limit to the backstop, and I think that would be a very good thing, it's got to fall before the next election".
She said the government needed to "get serious" about policy on a customs union and guarantee to "legislate for the protections around workers' rights".