During a working dinner with other EU leaders at a Brussels summit, May gave "a clear commitment that no EU citizen currently in the UK lawfully will be asked to leave the country at the point that the UK leaves the EU", a British government source said.
May set out what she said was a "fair and serious" offer on protecting the rights of an estimated three million Europeans living in Britain, whose futures have been thrown into doubt by its shock vote last year to leave the 28-nation bloc.
She did not set a cut-off date for those who are eligible for permanent residence and rejected a demand for the European Court of Justice to oversee the process and any resulting disputes.
In its position paper on EU rights published earlier this month, the European Commission said the court should have "full jurisdiction".
But May is determined to remove Britain from the ECJ's reach, saying that "taking back control" of its laws is one of the reasons Britons voted for Brexit.