At one point during the Oval Office negotiating session, it seemed President Biden and a half-dozen GOP senators chasing an infrastructure deal had landed on a workable figure: $1 trillion.

That number was dramatically scaled back from the White House’s initial $2.3 trillion vision for rebuilding the nation’s infrastructure, yet still far higher than the Senate Republicans’ opening bid. Some of the GOP senators in the room were unsure how they would reach the $1 trillion mark.

Then Sen. Roy Blunt (R-Mo.) asked Biden: Would it matter if that spending were spread out over five years, or eight? The president indicated to the senators that it wouldn’t.

The Republicans — some of whom had taken chocolate chip cookies, wrapped in a presidential seal, for the road — left the Oval Office believing that Biden would be satisfied with a framework that would spend $1 trillion over eight years and that it could include existing spending plans.

“I have had opportunities and dealings with him over the years, and he’s a straight shooter,” said Sen. Mike Crapo (R-Idaho), one of the six negotiators representing Senate Republicans with the White House. “If he gives you his commitment, you can count on it.”

What precisely Biden agreed to in the Oval Office has become a key factor in the ongoing negotiations between GOP senators and the White House, which it appears will continue beyond Memorial Day despite wide gulfs between the two sides concerning what to spend money on and how to pay for it.

Republicans this week began offering up details of the May 13 meeting after White House aides put forward a counteroffer that GOP senators said fell far short of what the president had pledged. The White House has declined to confirm details of the meeting but has repeatedly stressed that Biden himself signed off on the administration’s counteroffer. Five of the senators provided details about the meeting in interviews with The Washington Post and in other public statements.

The Oval Office session — one of several Biden has hosted as he hunts for an elusive infrastructure agreement — also provides insight into the negotiating style of the president, a creature of the clubby Senate who has landed significant bipartisan deals during his decades in Washington.

“I think it stems from his, kind of, innate Senate negotiating skills,” said Sen. Shelley Moore Capito (R-W.Va.), the point person for GOP senators in the infrastructure talks. In discussions, Capito recalled, Biden continually nudges both sides toward agreement, telling Republicans that the White House can “consider that” or “look at that” or that “I don’t object to that.”

In a CNBC interview Wednesday morning, Sen. Patrick J. Toomey (R-Pa.) said Biden implied to Republicans that a plan in the range of $1 trillion to $1.2 trillion would be acceptable.

“Look, I don’t want to try to reconstruct the conversation,” Toomey told The Post later that day. “But I can tell you, I distinctly had the impression from him that that was a ballpark that he thought he would be able to live with.”

Although the White House and Democratic leaders retain the option to pass an infrastructure package on a party-line basis, the discussions between Biden and GOP senators represent one of the president’s best hopes for a bipartisan policy achievement. The discussions between the White House and the GOP group — which includes Blunt, Capito, Crapo, Toomey and Sens. Roger Wicker (Miss.) and John Barrasso (Wyo.) — have been the most visible sign of Biden’s willingness to engage with Republicans.

Longtime friends and allies of the president say Biden has a distinctly nonantagonistic approach to negotiations.

“People think Joe is a good talker, and he’s a good talker. He’s also a good listener,” said Sen. Thomas R. Carper (D-Del.). “He never tries to put … somebody with whom he may have a disagreement on an issue, in an awkward position.”

That openness may be why Republicans were so taken aback by the $1.7 trillion counteroffer put forward by White House last week, a figure much higher than what they thought they had agreed to and one that included several provisions that GOP senators made clear were off the table for them, such as $400 billion for long-term care.

Biden sliced hundreds of billions in spending from his initial offer, although some of those cuts were priorities that already are moving in separate legislation set to pass the Senate in June.

In an interview, Capito said White House aides “very respectfully” pulled back from what Biden had agreed to with GOP senators, which prompted Republicans to push back harder with administration officials “because they couldn’t deny that this is what the president had said.”

“Here’s what I would say: I’ve been in situations myself, being the principal, where your staff’s in the room,” Capito said. “You’re preprogrammed to only go to step three, but you go to step four anyway. And then they have to come back and do a little … cleanup work.”

Moreover, Capito said she “fundamentally” believes Biden wants a deal with Republicans.

“I don’t want this to get into a ‘Biden doesn’t know what he’s doing.’ He totally understands,” added Blunt. “All of his training is as a senator who understands the importance of finding a place where everybody can be moving forward.”

Not wanting to give up just yet, Republicans decided at a huddle Tuesday morning in Capito’s office that they would continue the negotiations and send another counteroffer to the White House later in the week. At a news conference Thursday, GOP senators unveiled that counteroffer: $928 billion of spending over eight years, with $257 billion of that being new spending above and beyond what already exists.

The offer received a relatively warm reaction from the White House, which cited “several constructive additions” to the Republicans’ latest effort despite lingering concerns elsewhere in the plan.

“It is encouraging to see [Capito’s] group come forward with a substantially increased funding level — nearing $1 trillion,” said White House press secretary Jen Psaki, who notably used the more generous dollar figure to characterize the GOP proposal. “We appreciate the hard work that went into making this proposal and continuing these negotiations.”

Still, Biden himself began to indicate publicly that this negotiation, like others, would have an end.

“I told her we have to finish this really soon,” the president said before boarding Air Force One for a flight to Ohio on Thursday, shortly after he spoke with Capito on the phone. “We’re going to have to close this down soon.”

Biden felt the counteroffer was a productive step because Republicans have acknowledged that key programs need more funding, according to one person familiar with White House deliberations. The president plans to follow up after Memorial Day, although he knows that the negotiations cannot continue forever, the person said.

One important date for Biden that he is watching closely is June 9, when a House committee will begin working on a surface transportation bill that would form one pillar of the administration’s broader infrastructure plans, the person said.

The gap to be bridged in a steadily shrinking time frame is significant.

The White House continues to be concerned with how Republicans have proposed to pay for an infrastructure plan — by repurposing money from coronavirus relief packages that the administration says is simply not available — and with the exclusion from the GOP proposal of key priorities such as money to shore up veterans hospitals.

Republicans, in turn, say the White House continues to insist on spending that they think should not be included in a traditional infrastructure plan.

“It really does come down to: Is this middle-class Joe who was there in the meeting with us?” Barrasso said. “Or is this the $6 trillion man, now $7 trillion man . . . being pulled far to the left by Pelosi, Schumer and the group that really [is] promoting socialism? Which is what a lot of the spending is in the offer that we got back from them.”