As Congress barrels toward a government shutdown instigated by his own hard-right members, Speaker Kevin McCarthy is toiling to turn a fight over federal spending into a battle over President Biden’s handling of the border.

By floating a stopgap funding bill that would slash spending while imposing stringent immigration restrictions demanded by conservatives, Mr. McCarthy is attempting to shift the debate from an issue that has divided members of his party toward one where they are united — and where they believe Mr. Biden and his party are vulnerable.

“If they want to focus on Ukraine and not focus on the southern border, I think their priorities are backwards,” Mr. McCarthy said of Democrats at a news conference this week. In fact, it is not just Democrats but also Republicans in the Senate who have agreed to move forward with a temporary spending patch that would provide $6 billion in aid to Ukraine as well as $6 billion for disaster relief at home.

But Mr. McCarthy’s political play — along with broad agreement that the situation at the nation’s southern border has grown untenable — has spurred bipartisan talks on Capitol Hill about adding new immigration measures to government funding legislation. Some lawmakers now say it is likely that any deal to end what appears to be an inevitable shutdown on Saturday at midnight will eventually have to include some money or policy provisions to address the border.

Republicans believe the intersection of the funding crisis and a surge of migrants at the southern border provides new leverage to get additional border security money and changes in Biden administration policy either in a stopgap measure or ultimately in a deal to fund the government through the next year.

“We’re going to secure the border as part of any effort to keep the government open,” said Senator Lindsey Graham, Republican of South Carolina. “This is a problem that we’re going to address before we pass any permanent funding from the government.”

The pressure has started to influence some Democrats, who have expressed interest in adding border measures to a spending bill as a way of resolving the impasse, as Republicans dig in.

Senator Patty Murray of Washington, the chairwoman of the Appropriations Committee, acknowledged on Thursday that “there are colleagues concerned about doing more on border security — something I am willing to continue to discuss.”

“The border needs some attention, so I’d be open to it,” Senator Jon Tester, Democrat of Montana, told reporters this week.

Still, there is a vast gulf between what House Republicans are demanding and what the Senate will accept. G.O.P. hard-liners in the House are seeking to revive several Trump-era border policies, including the construction of a wall and the resumption of a “Remain in Mexico” policy of keeping migrants seeking asylum in detention centers or outside of the United States, as a condition for keeping the government open.

“It’s not something that’s passable,” Senator Kevin Cramer, Republican of North Dakota, told reporters of the House’s border provisions this week, even though he said he himself would be happy to vote for them.

As a shutdown nears, a small bipartisan group led by Senators Thom Tillis, Republican of North Carolina, and Kyrsten Sinema, independent of Arizona, has resumed its efforts to forge a broader immigration deal, meeting with lawmakers on the Appropriations Committee to discuss the possibility of adding such a package to a stopgap spending plan.

A group of Senate Republicans met behind closed doors on Thursday to discuss adding some border provisions to the Senate’s stopgap funding bill. The Senate voted overwhelmingly on Thursday to begin considering that spending patch, moving one step closer to passing it.

The House, on the other hand, was focused on a handful of individual yearlong spending bills, including one to fund the Pentagon, that would be dead on arrival in the Democratic-controlled Senate if Republican leaders managed to pass them. The military spending bill and other measures including homeland security funding narrowly passed late Thursday, largely on party lines. A bipartisan majority overwhelmingly approved legislation providing $300 million in aid to Ukraine, which had been stripped from the larger military bill at the insistence of hard-right Republicans.

In the Senate, G.O.P. senators were discussing adding $6 billion to their stopgap spending measure for securing the border, as well as a policy change that would raise the bar of eligibility for migrants seeking asylum. Republicans have long argued that many migrants abuse the current standard, which allows migrants with a “credible fear” of torture or persecution upon returning to their home country to make a claim.

“We’re trying to figure a way to grow the vote to avoid a shutdown, but also to have real teeth on the border and immigration reform that is credible and could potentially get done,” Senator John Cornyn, Republican of Texas, said on Thursday as he emerged from the meeting.

Mr. Cornyn added later: “I think given the nature of the border crisis, there’s a demand — and one I agree with — that actually this may present a golden opportunity for us to basically force the Biden administration to do its job and secure the border.”

Whether that would be enough to break the impasse and win over the far right in the House remains unclear. Beyond any border measures, any funding bill that passes the Senate would adhere to the spending levels agreed to by President Biden and congressional Republicans when they brokered the debt ceiling deal — levels that Mr. McCarthy and House Republicans have since reneged on and sought to cut even further.

And Representative Chip Roy of Texas, an influential conservative, suggested on Thursday that such efforts were futile, and that only Senate passage of the House’s stringent border bill — a near impossibility in the Democratic-controlled chamber — would persuade his colleagues to fund the government.

In a post on the X platform, formerly Twitter, on Thursday, he dismissed what he called “border security ‘gangs’ ” seeking a deal on border provisions. “The solution remains #HR2,” he wrote, referring to the House-passed legislation and calling on Senate Republicans to pass it as a first step toward “unlocking” government funding.

Kayla Guo contributed reporting.