By leaving the EU, Britain will be withdrawing from the bloc's policy of free movement of citizens among member states.
That leaves 3 million EU nationals in Britain, and 1 million Britons living in other member countries, in limbo.
Parliament's upper chamber, the House of Lords, plans to try today to pass a guarantee that EU citizens would be able to stay in Britain after Brexit.
Opposition peers hope to amend a bill authorizing the start of EU exit talks to include the promise, and have enlisted the support of some parliamentarians from the governing Conservatives.
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But the letter offers no guarantees of EU citizens' right to remain, saying only that the issue would be a top priority once formal exit talks start.
Prime Minister Theresa May plans to trigger Article 50 of the EU's key treaty, starting two years of exit negotiations, by March 31.
But she can't do that until Parliament passes legislation sanctioning the move.
If Lords amend the bill, it will have to go back to the House of Commons for another vote, delaying its passage and potentially threatening May's timetable.
A growing number of politicians and business groups are warning that the schedule already is tight. Former Prime Minister John Major warned Monday that the goal of agreeing on divorce terms within two years is "very, very optimistic."
The British Chambers of Commerce said yesterday that the break from the EU should be delayed if a post-Brexit trade deal remains incomplete after two years.
Foreign Secretary Boris Johnson told the group's conference that he wanted to go "full tilt and get it done within two years."
"I'm not saying there won't be some bumps in the road," Johnson added. But he said the UK and the bloc should be able to strike "a fantastic deal" that is good for British businesses.
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