“I’ve talked to a couple of key negotiators yesterday,” he said, “and they feel like they are making some progress, but I know Senator Schumer thinks there is going to be a deal cut behind closed doors and then jam it through the Senate and then jam the House. That’s not going to happen. But we are making some progress.”

The House has adjourned for the holiday, though Senate Majority Leader Chuck Schumer has kept the Senate in session. The negotiations on immigration are connected to legislation meant to provide supplemental aid to Ukraine, Israel and Taiwan.

“We do need to be aware of the fact that this is not just an exercise in the Senate,” Cornyn said. “It’s not just the Senate and the president agreeing to something. It’s something that can actually pass the House and be signed into law.”

Regardless, Cornyn said Republicans are still seeking to push the Biden administration to handle immigration better.

“What we are asking for is humane, orderly and legal immigration,” he said.

Speaking on NBC’s “Meet the Press,” Sen. Lindsey Graham (R-S.C.) also expressed doubts that a full agreement could be reached.

“The bottom line here is we feel that we’re being jammed. We’re not anywhere close to a deal. It’ll go into next year,” Graham said.

On the other hand, Sen. Joe Manchin (D-W.Va.) said he was “very encouraged” about the state of the negotiations.

“I’m very optimistic,” he said on CNN’s “State of the Union. “They’re moving in a very positive way. They understand that the border is broken. That glass ceiling has been broken. And we have got to stop this dangerous immigration that we have coming to our country from all over the world.”