President Biden’s efforts to finally make progress on a comprehensive overhaul of the nation’s immigration system have been stalled by Republicans, blocked by courts and rejected for violating arcane Senate rules.

Now, Democratic leaders are “seriously considering” a long-shot proposal intended to get around the political and procedural roadblocks by including language in the president’s sweeping social safety net package to provide temporary legal status to millions of undocumented immigrants, according to three congressional officials and several others familiar with the plans.

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The proposal — which drops one of the president’s key demands to create a direct path to citizenship for the immigrants — faces a string of challenges before it could even be formally considered for inclusion in Mr. Biden’s bill. But the fact that Democrats are still trying to wrestle it into the legislation is a sign of how desperate they are to fulfill promises they made to a critical segment of the party’s electoral coalition.

“This is a solution that gets us closer to our North Star of citizenship for 11 million,” said Kerri Talbot, the deputy director of the Immigration Hub, a pro-immigration group. “Lasting protections for 8 million people is nothing short of a breakthrough.”

Under the plan, the immigration measures would be included in the social safety net bill that Democrats intend to pass unilaterally through a fast-track process known as budget reconciliation, which allows certain spending and tax bills to pass by a simple majority vote. Those measures would expand the Homeland Security secretary’s authority to grant a temporary status known as parole to those who are undocumented and have lived in the United States for a decade or more, shielding them from deportation.

But the plan would have to pass muster with Elizabeth MacDonough, the Senate parliamentarian, who serves as the chamber’s arbiter of Senate rules that limit what can be included in Mr. Biden’s bill. She shot down previous attempts to include a path to citizenship in the reconciliation bill, saying the immigration plan was too vast and did not have enough of a direct impact on the budget.

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The legislation would offer undocumented immigrants not just protection from deportation, but also the ability to obtain a work permit — a point that immigration advocates say should more clearly connect the provision to the federal budget, making it easier for the parliamentarian to allow it under the rules of reconciliation.

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The Democrats’ plan would include most undocumented immigrants who entered the United States before Jan. 1, 2011, and could help between 7 million and 8 million people, the people familiar with the plan said. It would allow those covered to travel out of the country with the approval of Homeland Security.

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