(Bloomberg) -- Republican leader Mitch McConnell said significant issues remain in Senate negotiations on US-Mexico border restrictions demanded by his party in exchange for clearing assistance to Ukraine and other US allies.

McConnell and Senator James Lankford, the main Republican negotiator, told colleagues in a letter it isn’t clear whether the Senate will vote on the package this week, according to a person familiar with the matter.

Negotiators met over the weekend after Majority Leader Chuck Schumer delayed the Senate’s holiday break to allow for a possible vote this week. A vote would increase pressure on the House, which has already left Washington until January, to act early next month. 

“We’re not anywhere close to a deal,” Senator Lindsey Graham, a Republican from South Carolina, said earlier Sunday on NBC’s Meet the Press. “It’ll go into next year.”

A group of 15 Republican Senators urged the party’s conference chairman Senator John Barrasso not to hold a meeting until Jan. 8 when the House returns from its holiday recess, saying in a letter that “rushed” and “secret” negotiations with Democrats would not secure the border.

The move is an effort led by Senator Ron Johnson and other allies of House Speaker Mike Johnson, who has signaled that he prefers to move border talks forward in the new year.

With possible measures to curb migration across the border from Mexico dividing both parties, the two sides revealed little about the detailed state of play on Sunday, though some signaled optimism that the two parties will be able to reach a deal.

“I talked to a couple of key negotiators yesterday and they feel like they’re making some progress,” Senator John Cornyn, a Republican from Texas, said on Fox News Sunday. He described the negotiations “very delicate and difficult.”

Negotiators have made progress on asylum policy, which advocates of tighter rules say is being misused by economic migrants entering the US, Graham said. He said differences are wider on Republican efforts to limit President Joe Biden’s ability to grant parole to immigrants so they can live and work in the US while waiting for a ruling on their claim. 

Biden has offered changes to US border policy, and Homeland Security Secretary Alejandro Mayorkas met with negotiators in recent days as part of stepped-up engagement by the White House.

“They’re moving in a very positive way,” West Virginia Democrat Joe Manchin said on CNN’s State of the Union. “They understand that the border is broken.”

Republican Senator Thom Tillis has said the offer would give the president expanded authority to deport migrants once a certain number of undocumented migrants has crossed the border.

If the Senate reaches a deal, getting the House to agree to the package would be the next hurdle. 

Democrat Representative Ro Khanna said negotiations will have to include a broader range of lawmakers as some progressives and members of the Hispanic Caucus criticize proposals for stricter immigration policies. 

“You have to at least have someone like Nanette Barragan, the chair of the Hispanic Caucus, at the table,” Khanna said on Fox News Sunday. “You need to have some of the Latino senators at the table.”

Biden last week accused Republicans of holding Ukraine funding hostage in pursuit of “extreme” demands on border security. Ukrainian President Volodymyr Zelenskiy met Biden at the White House and addressed lawmakers in Washington last Tuesday but left empty-handed.

“All the eyes are on Congress right now, and we really hope there’s progress,” Oksana Markarova, Ukraine’s ambassador to the US, said on CBS’s Face the Nation.

--With assistance from Erik Wasson.

(Updates to include letter to Senator Barrasso in graphs 5 and 6.)

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