Rebel peers are confident of amending the Article 50 Bill with protections for EU nationals once the legislation arrives in the House of Lords.
The Bill, which grants Prime Minister Theresa May the power to trigger Brexit talks, was passed unamended by the House of Commons last week.
But both the Lib Dems and Labour have tabled amendments in the Lords offering protections for the rights of EU citizens.
A similar amendment was rejected in the Commons, but the makeup of the House of Lords means that opposition peers can unite to outnumber Conservatives in the chamber.
And sources within the Lib Dems say they have enough cross-party support to force through changes to the legislation.
Should they succeed, the amended bill would then be returned to the House of Commons for approval.
Although MPs would be able to force the Lords to reconsider, a long-running dispute could endanger the Prime Minister's hopes of beginning Brexit talks by the end of March.
May has repeatedly stressed her desire to offer similar guarantees to the rights of Europeans, but has been held back by an inability to secure equivalent protection for British nationals currently living on the continent.
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It comes as former Lib Dem peer Baroness Nicholson has warned the party away from "cheap political stunts" to delay Brexit.
"To my former colleagues I say this – don’t dishonour your democratic principles because you disagree with the outcome of the referendum. Don’t limit our country’s chances of success outside the EU in a cheap political stunt. Don’t wreck this bill," she said.