- May outlined five principles, notably that no EU citizen resident in Britain at a cut-off date would be deported.
- May also said those EU citizens who had lived in Britain for five years could stay for life.
- German Chancellor Angela Merkel called it a good start" but said many questions remained.
Theresa May offered fellow EU leaders a "fair" deal on Thursday for compatriots living in Britain after Brexit, though her peers sounded skeptical and demanded more detail from a prime minister weakened by an electoral misfire two weeks ago.
Given the floor for 10 minutes at the end of a Brussels summit dinner, her first since she launched the two-year withdrawal process in March, May outlined five principles, notably that no EU citizen resident in Britain at a cut-off date would be deported. There are roughly 3 million living there now.
That was, she told them, "a fair and serious offer", a British official said. It was "aimed at giving as much certainty as possible to citizens who have settled in the UK, building careers and lives, and contributing so much to our society".
Promising details on Monday, May also said those EU citizens who had lived in Britain for five years could stay for life.
Those there for less would be allowed to stay until they reach the five-year threshold for "settled status". Red tape for permanent residency would be cut there would be a two-year grace period to avoid "cliff edge" misfortunes.
German Chancellor Angela Merkel, who earlier said she wanted "far-reaching guarantees", described giving full rights to those in Britain for five years as "a good start" but said many questions remained.
"It is a first good step which we appreciate," said Austrian Chancellor Christian Kern. "Many details are left open. A lot of European citizens are concerned and not covered by May's proposal. There is a long, long way to go for negotiations."
Other leaders offered few details on their reservations.
May's push to set the cut-off date as early as March 29 this year, is unlikely to wash with many in the European Union, whose position is that nothing must change until Britain leaves -- scheduled for March 30, 2019. And there was much missing from an outline offer which the British previously called "generous".
Another sticking point could be May's rejection of another EU demand that expats be able to enforce their rights in the EU court. The source said they would have to accept British judges.
Brussels has been dismissive of May's call for sweeping and quick guarantees for expats, including over a million Britons on the continent, and says only detailed legal texts can reassure and take account of complex, multinational family situations.
Leaders had agreed with summit chair Donald Tusk not to open discussions with May and she left immediately afterwards, leaving the other 27 to discuss other Brexit issues without her.
They were briefed by Michel Barnier, who launched the Brexit negotiations for them on Monday, and discussed the move of two EU agencies from London after Britain quits.
Merkel and French President Emmanuel Macron, among others, had made clear that they did not want to be drawn into Brexit talks and wanted to focus on the future of the EU minus Britain.
May seemed keen to calm the mood with the continentals after weeks of sniping during her election campaign, stressing that London wanted a "special and deep partnership with our friends and allies in Europe".
Merkel also expressed a desire for constructive talks with Britain, but made clear that the EU's priority now was its own future: "We will conduct these talks in a good spirit," she said. "But the clear focus has to be on the future of the 27."
France's new president spoke of working with Germany to revive European integration and Macron did not refer at all to Britain during his remarks before talks got under way.
Weakened by an election she did not need to call, May has watered down her government's program to try to get it through parliament and set a softer tone in her approach to Brexit.
Yet her aims have held - she wants a clean break from the bloc, leaving the lucrative single market and customs union and so reducing immigration and ending EU courts' jurisdiction.
On Thursday, her finance minister, Philip Hammond called for an early agreement on transitional arrangements to ease uncertainty that he said was hurting business.
Reflecting confusion on the continent about what kind of Brexit she will ask for, Tusk said ahead of a separate meeting with May: "We can hear different predictions, coming from different people, about the possible outcome of these negotiations: hard Brexit, soft Brexit or no deal."
Some Britons had asked him if he could imagine Britain not leaving after all: "The European Union was built on dreams that seemed impossible to achieve.
So, who knows?," the former Polish prime minister said before quoting John Lennon's song "Imagine": "You may say I'm a dreamer, but I am not the only one."