WASHINGTON — Mitch McConnell, the Senate Republican leader and self-described “grim reaper” of liberal legislative dreams, settled into a routine of sorts during Barack Obama’s second term whenever he felt he was cornered by Democrats.

Mr. McConnell would rise from his chair in the Capitol, walk to his scheduler’s desk, smile a tight smile, and ask: “Can we get Joe Biden on the phone?”

That was precisely what happened in late 2012, when Republicans were still in the minority in the Senate, and Mr. McConnell hit an impasse with Harry Reid, the Democratic leader, over the elimination of Bush-era tax cuts for the wealthy.

As a New Year’s Eve deadline approached, Mr. Biden and Mr. McConnell hammered out an agreement in a dozen phone calls, aides to both men said, with Mr. Obama signing off on every move. The two sides struck a deal that delivered some, but far from all, of what Mr. Reid wanted. This year, as he runs for president, Mr. Biden cites that deal and others he cut with Mr. McConnell as proof of his skill in achieving bipartisan legislation in an otherwise hyperpartisan environment.