Republican negotiators are mulling a counterproposal from Democrats on a multibillion-dollar package of supplemental aid for disaster victims that would also handle a huge influx of migrants at the southern border.

Senate Appropriations Chairman Richard C. Shelby, R-Ala., declined to provide details on the offer, but said it could be the next step toward a bipartisan bill moving this week.

“We could, but we can’t budge on anything until they conclude and reach an agreement — first in principle and then they’ve got to put the language together,” Shelby said Wednesday morning. “Could they do it? There’s been a lot of hearsay running around on the Hill that we could be here until Friday.”

[Federal money hasn’t reached disaster victims]

Shelby declined to detail if the remaining issues are within the disaster aid portion of the package or if the disagreements are with the section that will appropriate billions of dollars to address the needs of migrants at the U.S.-Mexico border.

But he did hint that immigration policy, particularly provisions the White House wants in the package, could be causing problems.

“It’s dealing with the border, among other things, and what do you do with people. Do you detain them? Or do you turn them loose and never hear from them again?” Shelby said.

[House passes plus-upped disaster aid package]

Shelby said disagreements had “mostly” been ironed out over how much additional aid to provide to Puerto Rico to help it continue to recover from two deadly hurricanes in 2017.

Sen. Patrick J. Leahy, the top Democrat on the Senate Appropriations Committee, said he thinks a final agreement has to be reached Wednesday.

The Vermont Democrat said the Senate has late votes Wednesday in order “to try to keep everybody around to talk.”

“Shelby and I are pretty much agreed on what we want, but we’re waiting to see what the House wants. “ Leahy said he asked staff “to draft up the bill” because it is easy to make changes in that language.

Texas GOP Sen. John Cornyn said a vote on a disaster aid agreement is “possible” Wednesday evening, if there is agreement and consent.

“I don’t think they’ve got final text yet,” he said.

If a deal comes together, leaders in both chambers are still considering how the process would play out. Senate Minority Whip Richard J. Durbin, D-Ill., said it was possible the House votes first under expedited procedures before sending the bill to the Senate.

It’s not clear what the vehicle would be, though under that scenario it’s likely lawmakers would find some “shell” bill that’s previously passed in both chambers to replace with new text of a disaster and border aid package. That procedure would enable the Senate to bypass an initial cloture motion necessary to bring the measure up in that chamber.