WASHINGTON — Senator Chuck Schumer, Democrat of New York and the majority leader, is quietly considering trying to use a fast-track budget maneuver to legalize millions of undocumented immigrants should bipartisan talks on providing a pathway to citizenship fall apart.

Mr. Schumer has privately told members of the Congressional Hispanic Caucus in recent weeks that he is “actively exploring” whether it would be possible to attach a broad revision of immigration laws to President Biden’s infrastructure plan and pass it through a process known as budget reconciliation, according to two people briefed on his comments.

The move would allow the measures to pass the evenly divided Senate with a simple majority of 51 votes, shielding them from a filibuster and the 60-vote threshold for moving past one, which would otherwise require at least 10 Republican votes.

The strategy is part of a backup plan Mr. Schumer has lined up in the event that talks among 15 senators in both parties fail to yield a compromise. As the negotiations drag on with little agreement in sight, proponents are growing increasingly worried that Democrats may squander a rare opportunity to legalize broad swaths of the undocumented population while their party controls both chambers of Congress and the White House.

“Democrats must act,” says Sergio Gonzales, the director of the Immigration Hub, which pushes for a pro-immigrant agenda in Congress. “Now is the time. This year is the time. We must have citizenship this year.”

Mr. Biden’s immigration plan would provide a pathway to citizenship for an estimated 11 million undocumented immigrants, and increase diversity visas and border-security funding. But, conceding the long odds of achieving such extensive changes, lawmakers are focusing on cobbling together a package of smaller bills that would legalize about eight million or fewer undocumented immigrants.

They include House-passed legislation to grant legal status to people brought to the United States as children, known as Dreamers; immigrants who were granted Temporary Protected Status for humanitarian reasons; and close to one million farmworkers.

Speaker Nancy Pelosi last month endorsed the idea of using reconciliation to push through an immigration measure, citing the “budget impacts of immigration in our country.” Senator Patty Murray of Washington, the No. 3 Democrat, came out in favor of the approach last week.

Still, the strategy carries risks and is far from guaranteed to succeed.

Republicans involved in the talks warn that before Congress can act to address undocumented immigrants, it must address the large influx of migrants across the southwestern border. In March, border agents encountered nearly 19,000 children at the border — the largest number recorded in a single month — most of them fleeing poverty and violence in Central America, though the numbers are dropping.

“Before we can do anything meaningful on immigration, we’re going to have to deal with the current crisis at the border,” said Senator John Cornyn, Republican of Texas, who has been involved in the bipartisan talks. “I don’t think the public is going to tolerate us ignoring this crisis, and it’s just going to get worse unless we deal with it.”

As for moving Mr. Biden’s immigration agenda through reconciliation, he said: “I think they’re dreaming; I don’t think the parliamentarian will allow that. That’s not really the purpose of reconciliation.”

To pull it off, Democrats would have to grapple with strict budget rules that limit what can be done under reconciliation. They require that any policy change included must have a budgetary impact that is more than merely incidental. Other measures favored by liberal activists, such a federal minimum-wage increase to $15 per hour, have been nixed from a reconciliation package by the Senate parliamentarian, the ultimate arbiter of the rules, for failing to meet that bar.

The parliamentarian, Elizabeth MacDonough, recently said that the reconciliation process could be used at least once more during this fiscal year. That ruling was widely seen as paving the way for Democrats to advance Mr. Biden’s infrastructure bill using the fast-track process. It also set progressive activists to thinking about what else they could push through, including measures to address climate change, expand Medicare and revamp the immigration system.

A team of immigration activists and researchers as well as congressional aides is exploring the question, digging into the best way to present their case to Ms. MacDonough, who declined to comment for this article. They have found past precedents, including one from 2005, in which changes to immigration policy were allowed as part of a budget-reconciliation package, and they are tallying up the budgetary effects of the immigration proposals — which total in the tens of billions.

Researchers have dredged up supportive quotes from Republicans from 2005, when they won signoff for including a measure to recapture unused visas for high-skilled workers in a reconciliation package. Mr. Cornyn praised the move at the time as a way to “keep jobs here in America, rather than export them to places like India and China.”

The pro-immigration group FWD.us hired Kevin Kayes, a former assistant Senate parliamentarian, to help hone the procedural argument in favor of allowing the maneuver this year.

“Those provisions are the precedent for us,” said Kerri Talbot, the deputy director of the Immigration Hub. “A lot of things we’re trying to do now relate to what was approved in 2005.”

Ms. Talbot says she believes the total budget impact of the immigration bills under consideration is high enough to meet the reconciliation standard.

“We’re definitely in the tens of billions. We think we pass that test,” she said.

The estimated cost of the House-passed legalization measures is about $40 billion over 10 years.

Immigration advocates are also pushing for an expedited pathway to citizenship for the more than five million unauthorized immigrants who are essential workers, which is likely to carry an even higher budget impact on health care benefits, Medicaid spending and tax credits.

Twenty-two Democrats, including four senators, recently wrote a letter to Mr. Biden urging him to include an immigration overhaul in his infrastructure package. Many are worried that they will lose control of Congress in the 2022 midterm elections, and fearful that the Supreme Court will strike down former President Barack Obama’s protections for Dreamers.

“We ought to take this opportunity, this term, to finally do what the American people want us to do, which is to pass immigration reform,” Representative Joaquin Castro, Democrat of Texas and one of the letter’s authors, said in an interview. “I don’t think the Republicans should be able to run the clock out on this term before we pass meaningful immigration reform through the Senate.”

Yet not all Democrats are likely to support a unilateral approach. Senator Kyrsten Sinema, Democrat of Arizona, is co-sponsoring legislation with Mr. Cornyn to respond to the increase in migrants at the southern border by funding four regional processing centers in high-traffic Border Patrol sectors and improving the administration’s capacity to deal with such influxes.

For now, Senator Richard J. Durbin, the No. 2 Democrat who has for years pushed for a path to citizenship for the Dreamers, said that he was focused on passing a bipartisan immigration bill, and that Mr. Schumer had encouraged him to work to reach a deal with Republicans.

“We agree on a bipartisan basis we’ve got to reform the system,” Mr. Durbin said.

Senator Thom Tillis, Republican of North Carolina, another of the 15 senators involved in the immigration talks, said he believed that trying to use the reconciliation process for immigration reform would be a “disaster.”

But Mr. Tillis said he thought a bipartisan deal that couples a path to citizenship for the Dreamers with a greater investment in border security was still possible and perhaps getting closer.

“The crisis at the border is undisputable — even the president admits that now — so if we can work on that, and then work on some of the path options that I’ve supported in the past, I’m guardedly optimistic,” he said.