Theresa May is seeking an early Brexit deal for expats
May made the move in a meeting with the other leaders on Thursday in Brussels, the FT said.
There was no immediate response to May's request.
Irish Prime Minister Enda Kenny told reporters that May said she would "like to have the question of UK citizens living in Europe and European citizens living in the UK" resolved early on in the Brexit negotiations.
It is not the first time the prime minister has faced silence on the fate of expats and European citizens in the UK in the Brexit process.
Last month German Chancellor Angela Merkel and Donald Tusk, the European Council president, rejected an early deal on citizens' rights. Tusk said that the Brexit vote had created "anxiety and uncertainty" for Europeans living in Britain.
Similarly May has been attacked for not guaranteeing the rights of Europeans living in the UK.
Labour MP Gisela Stuart, who campaigned to Leave the European Union, said earlier this week that "Britain should make clear at the start of the Brexit negotiations that EU citizens already here before that date can stay.
"We should expect reciprocal deals for Britons living in European countries, but Britain should make the first move to demonstrate goodwill."