GOP senators said early Saturday morning that they've gotten a promise from Senate leadership to bring up a fix protecting young immigrants brought to the country illegally as children by early next month.

Asked if the agreement was to give a legislative fix for the Deferred Action for Childhood Arrivals (DACA) program a vote by Feb. 8, he added: "Yes, by Feb. 8 that vote would happen. We would start the process vote, invoke cloture on the Graham-Durbin bill and maybe another one."

He added that McConnell "remains committed to the ongoing, bipartisan, bicameral negotiations" but lawmakers must first reopen the government.

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"That being said, the first priority of the Congress must be to reopen the government and end the senseless Democrat filibuster of funding for the American military, American children’s health care and many other critical funding measures that our colleagues across the aisle," the aide said.

"As to how the leader would choose to process potential legislation on the floor if those negotiations stall is still under discussion, so I don't have anything new to announce," he said, referring to ongoing talks between the House, Senate and White House officials.

It's unclear if a deal to have an immigration vote by Feb. 8 would be enough to bring Democrats over to the three-week stopgap spending bill to fund the government, known as a continuing resolution (CR).

"That was something the majority leader didn't feel he could do. You can't bind the House in that way," Flake said.

Passing protections for those affected by Trump's decision to end DACA has emerged as the largest hurdle to passing a government funding bill or getting a larger budget deal.

Congress has until early March to pass a legislative fix, but Democrats have been adamant that it should be included in the legislation needed to prevent the current shutdown.

It's not the first time McConnell has promised to give an immigration bill a vote. Flake said in December that he would support the GOP tax bill in part because he got a commitment from the Republican leader to give a bill a vote by the end of January.

But McConnell said at the time that any bill would have to be able to win over the support of Trump, whose backing Republicans want before they tackle an issue so controversial to their base.

Flake said on Saturday that McConnell's new promise does not require Trump's approval, but other Republicans, including Sen. David Perdue (Ga.), could offer a conservative alternative bill.

"If there's a development here it's that Sen. McConnell realized we can't rely on the president to come up with agreement on a bill. We can't wait for the president," Flake said.

Asked if that was McConnell's message during a roughly two-hour vote, he added: "That was my effort and he agrees."

The "Gang of Six" bill would pair a DACA fix that includes a path to citizenship with roughly $2.7 billion in border security. It also would eliminate the State Department's diversity visa lottery, shift some of those visas to Temporary Protected Status and make narrow changes to family-based immigration.

But it's been panned by the administration, as well as conservatives in the House and Senate, as doing little to bolster the country's border security, boost interior enforcement or crack down on "chain migration," which allows citizens and permanent residents to sponsor family members.

Republicans in both the House and Senate have offered their own proposals more in line with a wide-ranging wish list released last year by the White House.

Perdue, who Flake mentioned could offer the alternative bill, said giving a DACA fix a vote was "being discussed."

"That's partly why we're going to be here tomorrow. And I would support that. I would support seeing both bills on the floor," he said.

Updated: 2:07 p.m.