BRUSSELS — Prime Minister Theresa May of Britain vowed on Thursday to protect the rights of millions of citizens of other European Union countries who are living in Britain after the country withdraws from the bloc.

Mrs. May made the pledge in remarks to the leaders of the other 27 European Union nations at a summit meeting in Brussels. It was intended to allay fears about mass dislocation stemming from Britain’s decision to withdraw.

“The U.K.’s position represents a fair and serious offer,” Mrs. May told the leaders over dinner at the summit talks. She said she wanted to give “as much certainty as possible to citizens who have settled in the U.K., building careers and lives and contributing so much to our society.”

More than three million citizens of other European Union countries live in Britain, while more than one million Britons live in the other 27 nations. Many of these people have formed families and raised children, and have been anxious about their status since the referendum a year ago when British voters decided to leave the union.

Angela Merkel, the German chancellor, described Mrs. May’s proposal as “a good start” although it met with skepticism from some other leaders. “There were so many details left open,” Christian Kern, the Austrian chancellor, told reporters.

A key issue is setting a cutoff date for European Union citizens living in Britain to qualify for what is called settled status, allowing them to remain indefinitely. The date must fall sometime between March 29, 2017, the day Mrs. May formally notified the union of its intention to withdraw, and the day the withdrawal actually takes effect — two years after the notice was given, barring an extension.

European Union officials have said they want Britain to guarantee their citizens’ rights to live in Britain right up to the date of withdrawal or Brexit as it is known. And they want the Court of Justice of the European Union to oversee residency and rights to education, health care, and pensions

But Mrs. May said on Thursday that the cutoff date had not yet been determined and that the system would be overseen exclusively by British courts, with no role for the European Court.

Five years’ residence is normally required to qualify to stay in Britain indefinitely, but Mrs. May offered on Thursday to allow a “grace period” of up to two years after the cutoff for people who had moved to Britain more recently.

Earlier in the day, several European leaders urged Britain to reverse course and remain a member of the European Union.

Donald Tusk, the president of the European Council, the body that organizes summit meetings, invoked the John Lennon song “Imagine” in discussing his hopes for a change of heart.

“Some of my British friends have even asked me whether Brexit could be reversed and whether I could imagine an outcome where the U.K. stays part of the E.U.,” Mr. Tusk told reporters. “I told them that, in fact, the European Union was built on dreams that seemed impossible to achieve. So, who knows? You may say I’m a dreamer, but I am not the only one.”

Mark Rutte, the prime minister of the Netherlands, had a far tougher message, saying that in order to avoid extensive economic damage, Britain needed to accept the jurisdiction of the European court and to go on allowing citizens of other European Union countries to live and work freely in Britain. “I hate Brexit from every angle,” said Mr. Rutte. But he added that the withdrawal was “a sovereign decision by the British people, and I can’t argue with democracy.”