But constructing a revised deal that might gain the support of a wider coalition of lawmakers, including some from the opposition Labour Party, has put Mrs. May on a political tightrope.

With her revamped plan, Mrs. May aimed to win over Labour lawmakers by offering the opportunity to vote on whether to put any plan for Brexit to a second referendum. Lawmakers would also be allowed to vote on whether to keep Britain, temporarily, in a type of customs union with the European Union that would eliminate tariffs and many checks on goods at borders.

But while the Labour leadership rejected the concessions as insufficient, many of Mrs. May’s pro-Brexit lawmakers were horrified, and several of those who had reluctantly supported the government in its last Brexit vote have said they would not do so again.

Another defeat would be particularly problematic since Mrs. May is not offering lawmakers a general vote on her deal but one on specific legislation to take Britain out of the bloc. Losing that would limit the options for a new prime minister, who would not be able to bring back the bill in the same parliamentary session.

On Wednesday, one Conservative lawmaker, Nicky Morgan, appealed to Mrs. May to reflect on whether to proceed with her Brexit vote, noting that the “consequences of it not being passed are very serious.”

Earlier, the environment secretary, Michael Gove, gave an agile and evasive interview to the BBC in which he hinted that the bill might not come forward as planned — without saying whether it would.

That question is connected to the timing of Mrs. May’s departure.

Earlier this year she promised to step down if lawmakers voted for her deal. Then, last week, she told senior Conservative lawmakers that, if her proposal was rejected again, she would set out a timetable for her successor to be chosen.