"I want to reassure all those EU citizens who are in the UK, who have made their lives and homes in the UK, that no one will have to leave. We won't be seeing families split apart," May said on arrival for the second day of a regular European Union summit in Brussels on Friday.
"Last night I was pleased to be able to set out what is a very fair and a very serious offer for EU citizens who are living in the United Kingdom," she said, adding that she would issue detailed proposals on Monday and seek reciprocal rights for about a million Britons living on the continent.
More detail was what most of the other 27 said they wanted, including German Chancellor Angela Merkel, who called it a "good start" but made clear that her focus was on the EU's future without Britain.
Belgian Prime Minister Charles Michel called it a "particularly vague proposal" and described it using a Flemish expression for a dubious gift.
"We don't want a cat in the bag," he said. "We want the rights of EU citizens to be permanently guaranteed."
In particular, the EU 27 want their citizens to be able to enforce their rights in Britain through the European Court of Justice, something May has flatly ruled out.
They also dispute her attempt to limit those rights potentially to people already living in Britain before she triggered Brexit three months ago.
Given the floor for 10 minutes at the end of a Brussels summit dinner, 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.
"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."
Brussels has been dismissive of May's call for sweeping and quick guarantees for expats and says only detailed legal texts can reassure people and take complex, multinational family situations into account.
Leaders had agreed with summit chair Donald Tusk not to open discussions with May and she left early on Thursday evening, 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.
Weakened by an election she did not need to call, May has watered down her government's programme 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.
(Additional reporting by Alastair Macdonald, Robin Emmott, Jan Strupczewski, Elizabeth Miles, Charlotte Steenackers, Philip Blenkinsop, Gabriela Baczynska, Tom Koerkemeier, Andreas Rinke, Jean-Baptiste Vey, Peter Maushagen and Alissa de Carbonnel in Brussels; Writing by Alastair Macdonald; Editing by Noah Barkin)