The EU's chief Brexit negotiator, Michel Barnier, says the UK must agree to "crystal-clear guarantees" to protect the rights of EU citizens.
The European Court of Justice (ECJ) will remain responsible for enforcing those rights, he said.
He demanded full protection of the rights of some 3.2 million EU citizens living in the UK and the 1.2 million Britons in other EU countries.
"No one should be confronted by a mountain of red tape," he warned.
Freedom of movement is one of the four essential freedoms that make up the single market, Mr Barnier said in a speech in Florence, and the integrity of the single market will "never, never be compromised".
"This point fell on deaf ears during the [Brexit] referendum campaign," he said. "We should not allow populists to take the political debate hostage."
Reassurance about the rights of EU citizens living in the UK, he said, must come first in the negotiations.
"I will not discuss our future relationship with the UK until the 27 member states are reassured that all citizens will be treated properly and humanely," he said.
"Otherwise, there can be no trust when it comes to constructing a new relationship with the UK."
Protection should apply for the lifetime of the citizens concerned, he added.
The rights of family members, including non-EU citizen family members, should continue to be protected, Mr Barnier said.
Brexit should not alter people's daily lives, he went on, and there must be equal treatment between all EU and UK nationals in the UK, as well the inverse.
The rights spelled out in the withdrawal agreement with the UK will need to be enforceable by the ECJ, he said.
Brexit negotiations are set to start next month after the UK's 8 June parliamentary election. The UK has two years to settle the terms of its exit from the EU.