BRUSSELS — If the British prime minister, Theresa May, still hopes to get Parliament to accept her withdrawal agreement from the European Union before a looming deadline this month, her chances lie with her attorney general, Geoffrey Cox.

Mr. Cox, who favors Brexit, was in Brussels on Tuesday evening to try to win binding assurances from the European Union that the border provision known as the Irish backstop will not last forever.

If Mr. Cox proclaims his satisfaction that the backstop is not meant to be perpetual and that Brussels does not intend to “trap” Britain into a customs union, then maybe — just maybe — Mrs. May can secure a parliamentary majority before the withdrawal date of March 29. Mrs. May has promised a vote by next Tuesday.

If Mr. Cox says that he is not satisfied, then it is likely that a divided British cabinet and Parliament will vote to ask the other member states of the European Union to extend the cutoff date for at least three months, an extension likely to be agreed upon at a key summit meeting on March 21-22. Such a delay would avoid an unregulated exit by Britain with unhappy though unclear consequences for both sides.