The Prime Minister made the comments as she updated the other 27 leaders on her Brexit plans.
The push for an early agreement on the status of citizens comes amid concern from many European leaders about the rights of their nationals in the UK after Brexit.
Irish Taoiseach Enda Kenny revealed the PM’s comments following the European Council summit meeting in Brussels, telling reporters Mrs May updated her fellow leaders on the Supreme Court case on Article 50 and her hope for a deal on EU nationals.
“She would like to have the question of UK citizens living in Europe and European citizens living in the UK dealt with in the early part of discussions that take place,” he said.
Mr Kenny’s comments came after Mrs May left the summit in Brussels without answering any questions on the UK’s break from the EU.
In a move signifying the UK’s direction towards the exit, the other 27 EU leaders had carried on their discussions without her in order to finalise their approach to the negotiations for the UK’s divorce from Brussels.
The national leaders agreed European Commission official Michel Barnier will be given the lead role for the EU, risking a turf war with MEPs who feel they should have a greater involvement in the Brexit talks.
European Council president Donald Tusk said the “short, informal meeting” had “reconfirmed our principles, meaning the indivisibility of the four freedoms, the balance of rights and obligations and the rule no negotiations without notification”.
Mr Tusk said representatives of the European Parliament would be invited to preparatory meetings but “maybe it’s not enough” to placate the angry MEPs.
Outgoing European Parliament president Martin Schulz has warned negotiations could be vetoed if MEPs are not fully involved, resulting in the “hardest Brexit possible”.
Meanwhile, Downing Street insisted the Brexit process could be completed within two years after the UK’s ambassador to the EU privately warned it could take a decade to finalise and even then may fail to be ratified by member states.
Number 10 said Sir Ivan Rogers was passing on the views of other EU nations.
“Ivan is there to report the views of others, he is doing the job of an ambassador,” a source said. “He was representing what others are saying to him.”
Downing Street indicated the Government believes it will be possible to complete both the “divorce deal” and a new trade agreement within the two-year timeframe set out under Article 50 of the EU treaties.
“We have been clear it’s a two-year process and we are not looking to extend it,” the source said.
Mrs May ducked questions about the potential bill the UK will be landed with as part of the divorce settlement with Brussels.
The issue will be one of the main items on the table in negotiations with the European Union, Czech Europe minister Tomas Prouza said. Reports have suggested the final cost could be as high as 60 billion euros (£50 billion), including payments to cover pension liabilities for EU staff.
Mr Prouza told Sky News: “This is what the UK has already committed to pay, and we would expect that the UK would honour its commitments. It will be one of the first issues coming up on the table.”
Downing Street said the UK would meet its obligations while it remained a member of the EU - but any financial settlement after that would be a matter for negotiation.