In a joint hearing organized by the committees of Civil Liberties, Employment and Petitions, MEPs said the EU should let go of the principle that "nothing is agreed until everything is agreed" in negotiations, because "a quick solution for citizens' rights is a matter of priority."
They said they were under a "moral duty" to ensure that EU citizens living and working in the UK would be able to retain their current rights under any Brexit deal.
"I strongly believe that when thinking about Brexit consequences, there is no greater concern than the fate of EU citizens who study, work and settle in the UK and also visitors, and of those British citizens who work or live in EU 27," said Renate Weber, Vice-Chair of the Employment Committee.
"Brexit will have a direct impact on the lives of hundreds of thousands of workers and students, on millions of tourists, as citizens' mobility and rights are at the heart of the European project. We have worked hard to guarantee citizens' uninterrupted access to all kind of benefits, and these rights should be safeguarded," Weber added.
In a statement, Liberal and Democrat MEPs said they were "determined that a mutual agreement on the rights of EU citizens in the UK and UK citizens in the EU will be a priority in the Brexit negotiations. ALDE MEPs will only give their consent for Brexit negotiations to move from phase one to phase two, when an adequate arrangement for citizens is in place."
Guy Verhofstadt, leader of the Liberals and Democrats in the European Parliament said:
"There are over 4 million EU citizens living in the UK or British citizens living in the EU, so the importance and scale of the issues to be resolved should not be underestimated by either side. We are determined to ensure that EU citizens in the UK can continue to lead their lives as they are today."
"In the forthcoming negotiations we will seek to protect and guarantee the rights of EU citizens in the UK on the basis of EU law, guaranteed by the Court of Justice of the European Union. I expect negotiations to be complex, but without an ambitious deal, it is unlikely the European Parliament will give its consent for negotiations to move on to discuss our future relationship. We have a lot of work to do to agree our divorce terms, before we can discuss the future and we intend to put citizens first, he said.