Mr. Biden views it differently. As the Senate prepared to pass the agreement on Thursday evening, he huddled with his chief of staff, Jeffrey D. Zients, along with Steve Ricchetti, counselor to the president, and other aides, in Mr. Zients’s office in the West Wing of the White House. Mr. Biden asked them what you might call a scorecard question: What percentage of Democrats in the House had voted for the deal, and what share were expected to in the Senate?

When Mr. Ricchetti told him the number of Democrats would be larger, in both chambers, than the share of Republicans supporting the deal, Mr. Biden was pleased. It was validation, in his view, that he had cut a good deal.

Mr. Zients referred to that vote share in an interview on Friday. “If you go back a few months ago, no one would have thought this was possible,” he said.

It was not an assured outcome. The negotiating teams came to the table with divergent views of the drivers of federal debt in recent years. White House negotiators blamed Republican tax cuts. Republicans blamed Mr. Biden’s economic agenda, including a debt-financed Covid relief bill in 2021 and a bipartisan infrastructure bill later that year.

The dispute occasionally grew profane. At one point, after Mr. Biden’s negotiators criticized the 2017 Republican tax cuts, a “very mild-mannered” aide to Mr. McCarthy stood up, shook his finger at the Biden team and hotly responded that their argument was nonsense, using a vulgarity, Mr. Graves recounted.