Labour will seek to force Mrs May into a decisive second Commons showdown on her Brexit deal by February 26.
The Prime Minister is instead offering MPs a further chance to vote on amendments to her Brexit strategy on February 27 in order to avoid a potential Cabinet split this week.
The Government is stepping up efforts to persuade the European Union to accept changes to the Irish border backstop in the Withdrawal Agreement.
Brexit Secretary Stephen Barclay is meeting EU chief negotiator Michel Barnier on Monday and Foreign Secretary Jeremy Hunt will visit Paris and Warsaw for talks this week.
On Thursday - Valentine’s Day - MPs will consider a motion on Brexit and a series of amendments, likely to include shadow Brexit secretary Sir Keir Starmer’s attempt to impose a new deadline for a vote on the deal.
But in an effort to see off attempts to bind the Government’s hands, Downing Street is promising another opportunity to table amendments - which are likely to include measures aimed at taking a no-deal Brexit off the table - on February 27.
The move is aimed at postponing a rebellion by ministers who are committed to removing the possibility of the UK crashing out of the EU without a deal on March 29.
But there is no commitment to hold a binding vote on the deal itself by the end of the month.
Communities Secretary James Brokenshire said: “I think that gives that sense of timetable, clarity and purpose on what we are doing with the EU - taking that work forward and our determination to get a deal - but equally knowing that role that Parliament very firmly has.”
He acknowledged that more work was needed to get the UK ready for Brexit on March 29, telling the BBC’s Andrew Marr Show there are “still steps that are currently being put in place” but “there is steady work that is going on, 10,000 civil servants that are now focused on this” and the Border Force was “ramping up” its staff.
Sir Keir earlier said his plan was necessary to put a “hard stop” to Mrs May “running down the clock” before the March 29 deadline.
He told the Sunday Times he fears the Prime Minister is “pretending to make progress” but actually intends to return to Parliament after the March 21/22 European Council summit the week before Brexit and offer MPs a “binary choice” - her deal or no deal.
“We can’t allow that to happen,” Sir Keir said. “There needs to be a day when Parliament says that’s it, enough is enough.”
Labour leader Jeremy Corbyn has written to the Prime Minister setting out five demands that would have to be met for his MPs to support a deal, including a permanent customs union and close alignment with the single market.
Deputy Labour leader Tom Watson told the Andrew Marr Show that if those conditions were not met then his party could move to supporting a second referendum.
“It seems to me we’re now at the point where we can have meaningful talks to get a deal between the main political party leaders,” he said.
“Or the only way to break the impasse is to have a public vote, and that remains our policy.”
Treasury Chief Secretary Liz Truss refused to rule out quitting if Mrs May did accept the demand for a customs union.
She told Sky News’ Sophy Ridge On Sunday: “I appreciate Jeremy Corbyn has come to the table but the reality is what he is proposing does not deliver on what we want as a country.”
Asked if she could stay in office if the Government backed a customs union she said: “I absolutely do not think that should be our policy.”