For Mrs. May, the summit was another ordeal after one of the most tumultuous five days in recent British politics. On Monday she abruptly postponed a Parliament vote on the Brexit deal, calculating that it would be defeated resoundingly. On Wednesday she survived a confidence motion among her own Conservative Party lawmakers, but more than one third of them voted to oust her, and she prevailed partly by promising that she would step aside before the next general election.

Now she faces twin, seemingly insurmountable obstacles: continuing to negotiate with Brussels for something she can call a new concession, while trying to allay British lawmakers’ fears of being tied forever to the European Union under a “backstop” agreement on the border between Northern Ireland, which is part of the United Kingdom, and Ireland, which is part of the European Union.

And she has to reschedule a vote in Parliament that she is widely expected to lose.

Her conundrum won little sympathy from the bloc’s leaders during the two-day summit meeting.

“When it comes to the future relationship, our British friends need to say what they want, rather than asking us what we want,” Mr. Juncker said. “And so we would like within a few weeks for our U.K. friends to set out their expectations for us, because this debate is sometimes nebulous and imprecise and I would like clarifications.”

An offended Mrs. May appeared to confront Mr. Juncker Friday morning in a testy exchange and, though video of the encounter did not provide audio, broadcasters confirmed the altercation with the help of lip readers.