European Council president Donald Tusk has said members will decide in November where EU agencies now based in Britain will be relocated.
No EU national currently living lawfully in the UK will be made to leave on the day of Brexit under proposals outlined by Theresa May to her EU counterparts.
Mrs May told a European Council summit in Brussels that she wanted to offer "certainty" to the estimated three million EU expats in the UK and ensure that families are not split up by Brexit.
But she made clear that the proposals would be adopted only if the same rights are granted to UK citizens living in the remaining 27 EU states in a reciprocal settlement.
And she set up a series of probable clashes with the European Union by suggesting she could set an early cut-off date for residency rights and rejecting a Brussels demand for the European Court of Justice (ECJ) to retain powers to enforce rights following Brexit.
Under Mrs May's plans, unveiled on the eve of the anniversary of the Brexit referendum, EU nationals who have lived in the UK for five years by a specific cut-off date will be given the chance to take up "settled status", granting them rights to stay in the country and receive healthcare, education, welfare and pensions as if they were British citizens.
Those resident for a shorter period will have the opportunity to stay on until they have reached the five-year threshold.
Those arriving after the cut-off date but before the date of Brexit will have a "grace period" - expected to be two years - within which to regularise their immigration status with a view to later seeking settled status.
The cut-off date is yet to be set, but will come between the day when Britain formally notified Brussels of its intention to quit on April 29 2017 and the day when it finally leaves, expected to be March 29 2019.
Mrs May also promised that the system will be streamlined, doing away with the 85-page permanent residency application form which has been the subject of loud complaints from EU expats.
It is thought that the UK is reserving the option of setting an early cut-off for residency rights in case there is a late surge of migrants arriving as Brexit approaches.
But the introduction of a "grace period" raises the possibility that large numbers arriving during withdrawal negotiations may be allowed to remain, at least for a few years.
And the outline deal leaves questions unanswered over whether individuals with settled status will be permitted to bring in children or spouses and whether the new status will be subject to conditions other than length of residency.
Further details are expected to be revealed in a paper to be published by the UK Government on Monday.
Speaking over dinner at the Brussels summit, Mrs May told leaders of the other 27 EU nations: "The UK's position represents a fair and serious offer and one aimed at giving as much certainty as possible to citizens who have settled in the UK, building careers and lives and contributing so much to our society."
She said the UK did not want anyone currently in Britain to be forced to leave.
But the proposals are likely to meet resistance in Brussels, which has already published its own formal proposals which would guarantee the rights enjoyed under EU law to any European expat resident in the UK on the date of Brexit.
Other than the different cut-off date, the key point of contention to be thrashed out in talks between Brexit Secretary David Davis and the EU's chief Brexit negotiator Michel Barnier is likely to be over the question of jurisdiction.
The EU proposals stipulate that the European Commission should have "full powers" to monitor and the ECJ "full jurisdiction" for as long as citizens' rights remain protected under the withdrawal agreement.
But Mrs May told her fellow leaders: "The commitment that we make to EU citizens will be enshrined in UK law and will be enforced through our highly respected courts."
And a senior British official added: "We have been clear on the ECJ that we are taking back control of our own laws."
Meanwhile, leaders of the 27 agreed a process which will see new homes on the continent agreed in November for two EU agencies currently based in London, the European Medicines Agency and European Banking Authority.