The House of Lords was due to vote on an amendment that inserts a promise to protect EU nationals' status into a bill authorizing the government to begin exit talks with the bloc.
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.
The government has said repeatedly that it plans to guarantee the right of EU citizens to remain in Britain, as long as UK nationals living elsewhere in the bloc get the same right.
The amendment commits the government to guaranteeing that EU citizens living legally in Britain when the bill is passed "continue to be treated in the same way with regards to their EU-derived rights."
Angela Smith, the opposition Labour Party leader in the Lords, said some EU citizens were "leaving the country now because they haven't got certainty."
Smith said adding a guarantee to the bill "not only gives clarity, it gives certainty."
The Conservative government does not have a majority in the House of Lords, and the amendment is backed by the main opposition parties, all but guaranteeing it will pass.
The House of Commons approved the bill earlier this month and the Lords is scrutinizing it this week. If the House of Lords amends the bill, it will have to go back to the House of Commons for another vote.
It's likely to be overturned there because the Conservatives have a majority, but the back-and-forth could delay passage of the legislation and potentially threaten May's timetable for starting EU exit talks.
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