Labour will seek to force Mrs May into a decisive second Commons showdown on her Brexit deal by February 26.
Last week, the passing of the Brady amendment, which called for "alternative arrangements" to the Irish backstop, paved the way for Theresa May to have another trip to Brussels in an attempt to renegotiate her Brexit deal that suffered a historic defeat in Parliament.
Speaking on the BBC's Andrew Marr Show on Sunday morning, Brokenshire said if a deal had not been agreed by February 27 then MPs would be allowed to again vote on their own proposals as to what should happen.
Britain is due to leave the European Union on March 29 but Parliament has rejected May's divorce deal, leaving the prime minister to seek changes from a resistant EU.
"A no-deal Brexit means a really hard border between north and south in Ireland, it's contrary to the Good Friday Agreement and it will cause an enormous fissure within the United Kingdom".
But there is no commitment to hold a binding vote on the deal itself by the end of the month.
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.More news: Microsoft pleads with companies to dump Internet Explorer
More news: Trump says no plan to meet China's Xi before trade deal deadline
More news: Acting U.S. attorney general snaps at Democrats over Russian Federation probe
British 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. It's got to know where we're heading before we leave'.
"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".
"Or the only way to break the impasse is to have a public vote, and that remains our policy".
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.
"You've got to decide which of those Brexits you want before we leave - otherwise, we're going to leave without clarity".
Asked by Sky News if she could stay in office if that became government policy, Tuss said: "I absolutely do not think that should be our policy".