Ministers say EU residents' status will be a priority once Brexit talks begin but opposition peers want a unilateral guarantee about their right to stay.
Lib Dems in the House of Lords say they expect a "clear victory" in a vote on an amendment to the Brexit bill later.
But Tory peer Lord Lawson said they were indulging in "virtue signalling".
Should the amendment be passed, it would be the government's first defeat on the bill - which will give Theresa May the authority to trigger Article 50 and begin official talks on the terms of the UK's exit from the EU.
If this happens, MPs could remove the Lords' proposed changes again when the bill moves back to the House of Commons later this month.
Home Secretary Amber Rudd, who was among ministers watching Wednesday's debate in the Lords chamber, has sought to reassure peers about the government's intentions, saying the 3.2 million EU nationals in the UK made a vital contribution and would be treated with the "utmost respect".
But she stopped short of offering a firm guarantee, saying this would not help the hundreds of thousands of UK citizens living on the continent as it could leave them in limbo if reciprocal assurances were not given by the 27 other member states.
The amendment, which would guarantee the rights of all EU citizens lawfully living in the UK at the time of the UK's departure from the EU, has cross-party backing.
Moving the amendment in the Lords, Labour peer Baroness Hayter said EU residents in the UK risked being used as "bargaining chips" in the negotiations and it was "in the gift" of the government to stop this happening.
While she was equally concerned about the two million Britons living on the continent, she insisted they should "not be traded against each other".
"These people need to know now, not in two years' time or even 12 months' time. They simply cannot put their lives on hold," Baroness Hayter said.
"Some are planning schools for their children, some are moving jobs, renting or buying a home or acting as carers. Some are receiving healthcare. Many more are working in our health service. All should have their uncertainty removed."
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