Mrs May's Brexit deal - which she spent months negotiating and had agreed with the European Union - covers the terms of the UK's divorce and the framework of future relations.
Housing minister James Brokenshire confirmed May's plans to the BBC: "The government will commit that the meaningful vote - the deal coming back - has not happened by 27 February, then we will allow a further motion, voteable in parliament, to take place, to give that sense of assurance as to the process moving forward".
The prime minister has to get an agreement with the European Union passed by MPs by 29 March to avoid a no-deal Brexit.
"We can't allow that to happen", Sir Keir told The Sunday Times.
Last week, Labour leader Jeremy Corbyn set out the conditions under which he would instruct his party to support an exit deal in parliament.
This would give them the chance to block a no-deal departure or make other interventions.
The Prime Minister has been in Brussels this week seeking concessions on the Northern Irish backstop, the reason many MPs rejected her vote by historic margins in January.
"No one could responsibly propose (a no-deal Brexit)".
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The prime minister wrote a conciliatory letter to opposition Labour Leader Jeremy Corbyn, after he proposed setting her a new deadline of February 26 for winning lawmakers' backing for a final exit deal.
"We would have a hard border, a very hard border, no-deal Brexit means a really hard border between the north and south of Ireland".
In early November 2018, former British PM Tony Blair called for a second referendum on Brexit, urging the country's parliament to turn down any divorce agreement between London and Brussels and let United Kingdom citizens decide as each option would have negative consequences.
The British government sought to win more time Sunday to secure European Union concessions that could pass parliament and avert a chaotic split from the bloc on March 29. "There needs to be a day when Parliament says that's it, enough is enough".
MPs voted to replace the controversial backstop with "alternative arrangements" in a vote on amendments to the PM's Brexit deal last month.
A number of government ministers will also be meeting their counterparts across the continent this week, in order to underline Mrs May's determination to achieve a deal. "It would be economically very, very unsafe for Britain and for the peace process in Northern Ireland, it would potentially be devastating", Blair claimed.
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.