WASHINGTON — 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.

They asked for anonymity to discuss the details of private negotiations.

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.

Those decisions have been cheered by Republicans, who have repeatedly criticized Democrats for trying to use an arcane budgetary maneuver to enact robust policy changes. Senator John Cornyn, Republican of Texas, has accused the Democrats of pushing for “a sweeping amnesty with no effort to control the growing immigration crisis and the southern border.”

But Democrats say they are still exploring what they can do to further the Biden agenda within the Byzantine rules of the Senate, including using the reconciliation process to bypass any Republican filibusters.

Democrats say they remain committed to finding a way to pursue a path to citizenship, but believe the latest option would be a chance to more quickly help many immigrants in light of the parliamentarian’s rulings. They note it is only one of several options they have for trying to make changes to the immigration system through reconciliation.

Significantly, the proposal would not put the millions of undocumented immigrants on a direct path to citizenship — the new measure alone would not entitle them to immediately get green cards. Immigrants currently eligible for a green card, such as the parents of adult U.S. citizens, would still be able to pursue citizenship under the plan.

Supporters of the plan have argued that the watered-down proposal should satisfy the parliamentarian, who had criticized the size of earlier versions.

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.

“This is a last-ditch effort to try and salvage something for the reconciliation process that can provide some level of protection for the undocumented while Democrats hold control of the Congress,” said Cris Ramón, an immigration consultant based in Washington, adding that it is “not necessarily I think where proponents wanted to land.”

The Democrats’s 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.

The proposal essentially codifies an enhanced version of the Deferred Action for Childhood Arrivals, or DACA, program introduced by President Barack Obama in 2012, giving work permits and legal status to people who came to the United States before 2012 and have lived in the country continuously since then.

“My guiding principle throughout the ongoing reconciliation process has been providing a pathway to citizenship for the largest swath of the undocumented community,” said Senator Robert Menendez, Democrat of New Jersey, who has worked on the issue, said in a statement. “In spite of the parliamentarian’s recent negative opinions, I will continue to work with my colleagues and a broad coalition of groups, including advocates, to find a path forward.”

The expansive social safety net legislation would expand Medicare to cover dental, vision and hearing, allow the federal government to negotiate the price of prescription drugs, and include funding for paid family leave, child care subsidies, free prekindergarten and community college, and investments to address climate change. Democratic leaders are currently working on paring down the $3.5 trillion package to draw the support of centrist Democrats needed to get the measure to Mr. Biden’s desk.

Michael D. Shear contributed reporting.