A panel has recommended European Union nationals living in the UK before the formal Brexit process is triggered should be allowed to remain in the country permanently.
Think tank British Future has claimed Theresa May's refusal to agree the status of the nearly three million Europeans in the country unless the rights of Britons living on the continent are guaranteed is "morally wrong".
It also called for the UK to demonstrate "goodwill" by "making the first move" ahead of negotiations.
The panel added all 2.8 million EU nationals should be eligible for permanent residence with the same health, social and education rights as British citizens.
Labour's Gisela Stuart, a leading Leave campaigner who headed the inquiry, said: "Our inquiry found that people from different sides of the referendum and politics can quite easily agree on practical, straightforward ways to ensure that EU nationals can stay in Britain with their rights protected.
"We determined that the triggering of Article 50 should be the cut-off date, after which EU citizens moving to the UK would not be entitled to stay permanently after Brexit. This would limit any 'pull factor' for EU citizens not already here in the UK.
"Britain should make clear at the start of the Brexit negotiations that EU citizens already here before the triggering date can stay. This would send a clear signal about the kind of country the UK will be after Brexit and the relationship we want with Europe.
"We should expect reciprocal deals for Britons living in European countries, but Britain should make the first move to demonstrate goodwill.
"That's the right thing to do and something that most voters agree with too."
A total of 1.2 million British nationals live in other EU countries.
Owen Tudor, TUC head of EU and international relations, said: "EU nationals working in the UK and making a contribution are anxious about what happens to them when Britain leaves the EU. They need to know about their future rights to live and work here.
"Everyone agrees this needs to be resolved quickly and fairly. This shouldn't be up for negotiation.
"The Prime Minister should make the first move to unblock this ghastly uncertainty. It is morally right and pragmatically sensible."
A Government spokesman said: "The Prime Minster and other ministers have been absolutely clear that they want to protect the status of EU nationals already living here.
"The only circumstances in which that wouldn't be possible is if British citizens' rights in European member states were not protected in return."