"We made significant progress," White House counselor Steve Richetti told ABC News.
Three Senate negotiators -- Republicans Susan Collins of Maine, Ohio's Rob Portman and Montana Democrat Jon Tester -- clarified to reporters that there were still some details to be worked out on how the $1.2 trillion package is to be paid for, though all agreed that a deal was at hand.
"There is a framework of agreement on a bipartisan infrastructure package. There's still details to be worked out," said Collins.
"For the most part we have a framework, but there are components within that framework that need to be worked out," Tester said. "Do you think we'd be going to the White House tomorrow just to drink tea?"
When asked if the group of 21 senators had a deal, Portman said, "I wouldn't use exactly those words, but I would say that we're very, very close, and we're going to now do the outreach. It is important to grow the vote from the middle out."
Portman was referencing the crucial sales job now underway with senators outside the group of 21.
The core, bipartisan group of 10 senators - five Democrat, five Republican - has been at this effort for a month.
Senior White House aides now must show the president the final numbers.
At the White House Wednesday afternoon, Biden was asked how he feels about the bipartisan infrastructure deal and he gave the first hint that the negotiations were nearing completion.
"I'll tell you that when I get the final numbers tonight," he said.
The mood in the Capitol Wednesday night was light and festive with pizza brought in near the end of the last round of talks.
Lawmakers are aiming to have staff wrap up the final details before senators leave for a two-week recess.
"I think it's really important that before we go home, we get the details ironed out, but considering where we started and where we are -- both sides negotiated in good faith, and this is important. It's important not only to the functioning of the Senate, it's important to America, and it's important globally to show that the Senate can work together," said Collins.
Senators had been at odds for weeks over how to pay for the package, which includes $579 billion in new spending, with Republicans refusing any tax cuts and the White House rejecting any fees or tax increases for Americans making under $400,000 annually.
But while this is a significant victory with substantial momentum, it remains to be seen how progressives will receive the plan which is far less than they had hoped for.
Democrats are also simultaneously crafting a sweeping, filibuster-proof plan to wrap in as much as $6 trillion in spending on "human infrastructure," which includes elder and childcare, climate change, Medicare changes and immigration reforms. That package would, in part, be paid for by raising taxes on the wealthy.
It remains to be seen if more centrist Democrats who are part of the bipartisan infrastructure talks can support such a sweeping package. Some progressives want to secure their agreement on the larger package -- which would need just 50 votes by using a budget procedure known as reconciliation -- before they would support any bipartisan deal.
Senate Democratic leaders have a difficult road ahead to get both packages passed Congress this year. Both House Speaker Nancy Pelosi and Senate Majority Leader Chuck Schumer have said they want these infrastructure packages on their respective floors in July.
ABC News' Molly Nagle contributed to this report.