Senate Republicans will give President Biden a revised infrastructure offer next week after a sit-down at the White House on Thursday.

Biden met with a group of 10 Republicans, led by Sen. Shelley Moore Capito (W.Va.), where they discussed two of the biggest sticking points: what qualifies as infrastructure and how to pay for it.

"He asked that we would come back with another offer, with more granularity to it and more details, and so we agreed to do that," Capito said.

Republicans are planning to give Biden their revised counteroffer early next week, and didn't rule out meeting again with Biden.

"He wanted it pretty quickly," Capito said. "I made it clear this was not a stagnant offer from us, and I didn't want it to be perceived that way."

Sen. Roy Blunt (R-Mo.) said that he also expected Republicans to come back with "more specifics" early next week on "on our idea to pay for things and further the list that we talked about with him."

"We were pretty far down the road," Blunt, a member of GOP leadership, said about Thursday's meeting.

The GOP group unveiled a proposal earlier this year that offered $568 billion for infrastructure — dramatically smaller than Biden's $2.3 trillion jobs plan. Capito, on Thursday, didn't rule out that Republicans would go higher in their next pitch that they give to the White House.

"Maybe some different numbers too," Capito said.

The next GOP proposal is likely to remain focused on "core" infrastructure including roads, bridges, water and broadband, though Capito said Republicans needed to lock down the areas they would focus on.

"I think we're still looking for a bill that would be traditional infrastructure with some flexibility. Broadband would be something that wouldn't have been traditional that I think we would argue could be now," Blunt said.

Biden discussed an area of interests for him: electric cars and competing with China. Republicans also discussed using public-private partnerships to help cover the costs instead of undoing the 2017 tax bill, viewed as a red-line for their party.

"I think what he wants to see is 'OK, I get it you don't want me to touch the 2017 taxes ... well then how are we going to pay for this,' " Capito said. "The [Paycheck Protection Program] was part of the discussion but there were other things discussed."

Both Biden and Republicans sounded relatively optimistic about the chances of getting a deal on infrastructure.

"We'll see if we can work out some, on a compromise on infrastructure. And I know they're sincere about it, so am I,” Biden told reporters who were briefly allowed into the meeting Thursday afternoon.

Sen. Mike Crapo (R-Idaho), who took part in Thursday's meeting, said that he left the meeting "with solid optimism."

The positive notes from the GOP senators on Thursday comes after Senate GOP Leader Mitch McConnell (Ky.) also sounded optimistic about the chances of getting a deal after his meeting with Biden on Wednesday.

"I think both sides would like to get an outcome," McConnell said during an interview with Fox News, adding that they "discussed an issue upon which there's a great chance we can get a bipartisan outcome."

Republicans say they are negotiating with Biden fully aware that Democrats could still try to pass his broader definition of infrastructure that doesn't get bipartisan support — like education, home care and child care — through reconciliation, which allows Democrats to bypass the 60-vote filibuster.

Biden indicated in an MSNBC interview on Wednesday that he was interested in passing what gets GOP votes in one package and trying to go it alone for the rest.

"So I want to know, what can we agree on? And let’s see if we can get an agreement to kick-start this. And then fight over what’s left and see if I can get it done without Republicans, if need be," Biden said.

GOP senators involved in the negotiations say they would be willing to cut a bipartisan deal with Biden on infrastructure even if Democrats were going to try to push through the rest of the president's plan with a simple majority.

"Why wouldn't we work with the president of the United States? ... We know that they have that option. We used that option in 2017," Capito said, referring to the 2017 tax bill.

Blunt added that "there's no sense on our part that if we all agree on this infrastructure package, that you wouldn't try to do the rest of it some other way."