"We need to give assurances to our citizens at an early stage of negotiations that they will have a right to stay [in whichever country they are living]," May said at a summit in Brussels this week, according to a note of proceedings seen by the Financial Times.
There was no immediate response from any other state, and May left the summit shortly afterwards, the Financial Times said.
Sweden's prime minister Stefan Lofven later said that May's statement was "positive" although others were more dismissive, the newspaper said.
"It’s a clear ambition that Britain does not want to complicate life for all these people who actually live in Great Britain," Lofven said.
Employment law expert Euan Smith of Pinsent Masons, the law firm behind Out-Law.com, said: "This would be a very welcome step towards getting clarity for EU employees in the UK, who have been anxious about their future and who are unsure as to whether or not they will be permitted to stay. Likewise, if other EU states are agreeable to the proposal, it will give some comfort to UK nationals who are employed in EU countries that they may not have to leave. This is just the first step however. There may well be further political wrangling about this proposal in the months to come."