Senate Minority Leader Mitch McConnell said that "we still have amendments that need to be processed. Once they are we’ll be able to wind things down."

Several conservative senators have signed off on wrapping up as soon as possible, including Rand Paul of Kentucky, Mike Braun of Indiana and Mike Lee of Utah. Paul said the bill's finances are questionable and he won't support it, but "I'm fine to finish whenever people get tired of whatever we're doing right now."

It's also possible that Senate Majority Leader Chuck Schumer moves to shut off debate on the bipartisan infrastructure bill as soon as Thursday afternoon, which would set up a crucial vote this weekend on the deal. Speeding things up would require agreement from all 100 senators.

Many Republicans had been waiting for the Congressional Budget Office score, which dropped on Thursday afternoon. That score showed that the bill's mix of $550 billion in new spending and the financing to pay for it would increase deficits by $256 billion over 10 years.

GOP leaders huddled on Thursday afternoon to discuss the path forward before a private party meeting of all 50 Republicans senators. Lee, the Steering Committee chair, is involved in the discussions.

"Everybody understands that right behind this is going to be the budget. And I don't think anybody is looking to extend this out any longer than necessary," said Sen. John Cornyn (R-Texas), who also attended the meeting in McConnell's office.

Failing that agreement, Schumer may put things in motion Thursday after the Senate considers more amendments to the bill, though no final decision has been made. That would set up a Saturday vote to overcome a filibuster, when the bill would need the support of at least 10 GOP senators.

Once the bill beats a filibuster, it will be on a glide path to passage. Immediately after that, the Senate will turn to consider a Democratic budget that will set up potential future passage of a spending bill as large as $3.5 trillion. Senate leaders are discussing the particulars of how they can wrap up both issues but allow rank-and-file members sufficient time to debate and consider them.

Schumer did not indicate his intent to imminently wrap the infrastructure bill up during Thursday morning floor remarks. But he defended his stewardship of the process and boasted that under his control the chamber has considered more amendments so far than during the GOP-run years of 2019 and 2020 combined. He also signaled he wants to finish the bill as soon as he can.

“Today we will consider even more amendments. And then hopefully we can bring this bill to a close very shortly,” Schumer said. “Our goal is to pass both a bipartisan infrastructure bill and a budget resolution during this work period, and we will stay here to get both done.”

As of Thursday morning, the chamber had considered 22 amendments, the majority of them from the GOP. Republicans are privately signaling they are likely to put up the votes to advance the bill, provided Thursday’s amendment consideration goes smoothly.

"I literally don't know [what will happen]. I know what we're aspiring to do but there's no agreement yet," said Senate Majority Whip Dick Durbin (D-Ill.). "The general notion is the sooner the better that we can move to the budget resolution."

Democrats plan to move the infrastructure bill toward a final vote as soon as the bill’s GOP supporters are satisfied with the opportunity to offer amendments. Thus far, 18 GOP senators — including McConnell — have signaled support for the infrastructure bill on key procedural votes.

McConnell warned Schumer earlier this week not to cut off amendment consideration, urging Schumer to go “slow and steady.” But the consideration of the long-awaited infrastructure bill may be coming to a natural close, with many Republicans expected to attend the Friday funeral of former Sen. Mike Enzi in Wyoming and then fly back for Saturday work.

Marianne LeVine contributed to this report.

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