Under the UK's plans, anyone who arrives in the country before it leaves the EU will be allowed to apply for permanent "settled status".
They must have lived or go on to live in the UK for five years to receive it.
But Theresa May wants to change the rights of those who arrive after the official divorce date - expected to be 29 March 2019 - so they would be subject to the new post-Brexit immigration scheme.
That would likely mean visas, registration on arrival and an ability to add caps on benefits to EU migrants.
Brussels insists otherwise - saying freedom of movement should continue until the transition ends when Britain formally quits the customs union.
Guy Verhofstadt, the European Parliament's Brexit co-ordinator, said the UK had to remain subject to EU legislation and case law for the transition to work.
He wrote on Twitter: "Citizens' rights during the transition are not negotiable.
"We will not accept that there are two sets of rights for EU citizens. For the transition to work, it must mean a continuation of the existing acquis with no exceptions."
The fight promises to build over the next two months, as both sides hope to reach a deal by the EU Council summit at the end of March.
Brexit Secretary David Davis said on Thursday he was "fairly confident" the deadline would be met.
It comes after Theresa May tried to assuage the fears of her Brexiteer MPs by rejecting some of the demands from the EU's chief negotiator Michel Barnier.
Speaking on a trade trip to China, the PM said: "I'm clear there's a difference between those people who came prior to us leaving and those who will come when they know the UK is no longer a member of the EU.
"When we agreed the citizens' rights deal in December, we did so on the basis that people who had come to the UK when we were a member of the EU had set up certain expectations.
"They made a life choice and set up certain expectations and it was right that we have made an agreement that ensured they could continue their life in the way they had wanted to.
"Now for those who come after March 2019, that will be different - because they will be coming to a UK that they know will be outside the EU".
Meanwhile, a Scottish minister has promised to publish a UK paper on the impact of Brexit if he is given a copy.
Mike Russell said the document, a draft of which was leaked earlier this week, "needs to be published".
He has written to Mr Davis and urged him to share the analysis with the Scottish government.