WASHINGTON — The top two senators negotiating an immigration deal cautioned Monday that the clock is ticking and they may not be able to reach an agreement this week before the Senate is scheduled to adjourn for the rest of the year.

That means President Joe Biden’s package of aid to Ukraine and Israel could falter with it, with the debate slipping into an election year that will make it even harder to secure a bipartisan agreement.

Sen. James Lankford, R-Okla., who is leading the discussions for Republicans, said he still believes negotiators are “making progress” but cautioned that “every day” that goes by without a deal “makes it more complicated” to reach one.

“There’s no way to get it done this week,” Lankford said in an interview. “The question is are we staying in next week or does this actually move into early January to be able to resolve? That’s a big unknown at this point, and that depends on how the negotiation actually goes and how we’re actually working through to be able to actually get text that actually works.”

Sen. Chris Murphy, D-Conn., a Biden ally who is leading the negotiations for Democrats, has voiced frustration with GOP demands, calling them extreme and implausible. He said there “can be a path” if Republicans revise their approach based on what can realistically pass Congress with bipartisan support.

“Obviously, we don’t have the benefit of time right now,” he said.

Asked whether the Senate can finish a deal this week, Murphy said: “I’ve seen stranger things happen in the Senate. But that certainly seems like an uphill climb — but not impossible.”

Ukrainian President Volodymyr Zelenskyy is expected Tuesday to visit Capitol Hill to speak to senators and meet with Speaker Mike Johnson, R-La., who has endorsed the GOP demands to block Ukraine aid unless Congress pairs it with tougher immigration and asylum laws.

“He’s preaching to the converted, as far as I’m concerned. So I don’t know what this is designed to do,” said Sen. John Cornyn, R-Texas. “I don’t see what the purpose is, but obviously we’d be glad to listen to what he has to say.”

Senate Republicans proved last week that they were unwilling to debate a bill to grant aid to Ukraine and Israel without immigration restrictions, as all 49 of them filibustered the package. The failed vote Wednesday rekindled negotiations that had previously faltered. Murphy and Lankford huddled with Sen. Kyrsten Sinema, I-Ariz., on Thursday for about an hour, a source said.

But there has been no breakthrough in the negotiations, senators said.

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Lankford said he’s happy to hear what Zelenskyy has to say, “but hearing their national security issues also reminds us of our national security issues. Quite frankly, we’re not going to go help other countries and not look at actually what’s happening in the United States.”

Sources with knowledge of the negotiations say Democrats have been willing to make concessions in tightening asylum rules, but the talks have been stuck over GOP demands to curtail presidential powers to use “parole” to temporarily admit asylum-seekers. Democrats fear the proposals to reform parole would force mass detentions and deportations that they consider cruel.

The House and the Senate are scheduled to leave town for the holidays in just three days, leaving precious little time to resolve a series of thorny issues that have bedeviled Congress for years. They could cancel some or all of recess to keep working on it, but that’s unlikely to happen unless they are close to an agreement they can vote on.

Even if the Democratic-led Senate reaches a deal, there’s no guarantee it will be passed by the Republican-controlled House, where conservative hard-liners are making demands that Democrats roundly say are nonstarters.

“Hopefully the House is willing to stay, if we can get a deal done,” Murphy said. “Hopefully the House is willing to stay and process anything the Senate can move, but that’s their own decision.”

Sen. Joe Manchin, D-W.Va., said Congress should stay in town until it reaches a deal.

“I want to see a reason why we should leave. That means that basically they don’t have the concerns that we have with Ukraine,” he said, referring to the country’s ability to defend itself against Russia. “I still have, very much, those concerns. We should stay and get it done.”

Sen. Lindsey Graham, R-S.C., dismissed the need for a briefing on Ukraine from Zelenskyy and said senators are well aware of the situation. “I think it hurts, they’re dragging him into a domestic political debate,” Graham said. “If they had asked me I would have said you shouldn’t do this.”

Sen. Joni Ernst, R-Iowa, said she also wants to hear a plan from Zelenskyy for “how they intend to win this war” and what Ukraine needs from the U.S. to do so.

“And then, as far as moving forward on the package, it has to have border policy, and the president knows that,” she said. “I’ve had so many Democrats, both the House and the Senate, that have said, ‘This would make our election a lot easier if the president will just move forward on border policy.’ So there are a lot of folks that really think we need to get border policy done.”