Within days, plans were set for two planes to take off — one from Moscow, where Ms. Griner had been transferred, and another from the United States, with Mr. Bout.

One question to resolve: Where to make the swap? In the Cold War days and even as recently as a Russian-American spy swap in 2010 under President Barack Obama, prisoners were exchanged in the middle of Europe: the Glienicke Bridge in Potsdam, Germany, as made famous in the Tom Hanks movie “Bridge of Spies,” or in Vienna, as with the 2010 trade.

But because of American and European sanctions imposed on Russia after its invasion of Ukraine in February, Moscow was not willing to send a plane anywhere in Europe for fear that it could be seized. Even long-neutral Switzerland had joined sanctions against Russia while Helsinki, Finland — a prime Russian-American meeting site during the Cold War — was no longer acceptable because the country is joining NATO.

The compromise became the United Arab Emirates, a small Gulf state that is friendly with both Washington and Moscow. Sheikh Mohammed bin Zayed Al Nahyan, the president of the Emirates, had raised Ms. Griner’s case with Mr. Putin during a meeting in October, and so the Emiratis were happy to facilitate the transfer. The arrangements were made: Both sides would send planes to Abu Dhabi, the capital.

Cherelle Griner, Ms. Griner’s wife, was invited to the White House on Thursday morning ostensibly to meet with Mr. Sullivan for an update. When she arrived, however, Mr. Sullivan surprised her by taking her to the Oval Office, where Mr. Biden broke the news that Ms. Griner was coming home. At that point, Ms. Griner was on the ground in Abu Dhabi, while Mr. Bout’s plane was 30 minutes out.

Once he landed, his clemency document was finalized and the swap proceeded, captured in grainy video distributed by Russian state media. It showed Ms. Griner and Mr. Bout being walked to the middle of a dusty tarmac, escorted by officials of their countries. After a brief stop, Ms. Griner was led away in one direction, while Mr. Bout left with Russian officials in another.

Mr. Biden and Cherelle Griner celebrated before cameras in the Oval Office. But once the journalists were ushered out, the president had another, grimmer task: He had to call Mr. Whelan’s sister to explain why he was not coming home, at least not yet.

Neil MacFarquhar contributed reporting from Paris, and Anton Troianovski from Moscow.