Communities Secretary James Brokenshire said Parliament would get to pass judgment on May's Brexit plan "no later than Feb 27".
"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".
Brexit Secretary Stephen Barclay is meeting European Union chief negotiator Michel Barnier on Monday and Foreign Secretary Jeremy Hunt will visit Paris and Warsaw for talks this week.
Critics of the current plans 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.
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May wants to win over lawmakers in her Conservative Party with changes relating to the Northern Irish border, but the EU has refused to reopen that part of the deal and instead wants May to pursue a compromise with the main opposition Labour Party by agreeing closer UK-EU ties.
In an interview with Sky News, Mr Blair said: "No one could responsibly propose this [a no-deal Brexit]".
"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".
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.
He said: "We have got to put a hard stop into this running down the clock".
Reaching a deal is "profoundly" in the interest of the U.K., Brokenshire told BBC Television's "Andrew Marr Show", but it is also important to give a sense of clarity and goal, which he believes May's new vote pledge will do. "And that's what we want to do this week".
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