"I'm glad to say Brittney's in good spirits... she needs time and space to recover," president Joe Biden said at the White House.

According to Russian media reports, Bout, also known as the "merchant of death," has arrived in Moscow.

"In the middle of the night they simply woke me up and said 'Get your things together' and that was it," Bout said in brief remarks to a reporter after landing in Russia.

Griner was arrested at a Moscow airport in February 2022 for possessing cannabis oil and sent to a penal colony last month.

The Biden administration proposed a prisoner exchange in July, aware that Moscow had long sought Bout's release.

The elaborate swap involved two private planes bringing the pair to Abu Dhabi airport from Moscow and Washington, and then flying them home.

Footage on Russian state media showed them crossing the tarmac with their respective teams.

Speaking in the Oval Office, Brittney Griner's wife Cherelle praised the efforts of the Biden administration in securing her release: "I'm just standing here overwhelmed with emotions."

According to a joint Saudi-UAE statement, Saudi Crown Prince Mohammed bin Salman played a leading role in mediation efforts, along with UAE President Mohammed bin Zayed al-Nahyan.

The heir to the Saudi throne has good relations with Russia's Vladimir Putin and in September he helped coordinate a complex swap of hundreds of prisoners held by Russia and Ukraine.

But the White House denied that any mediation had been involved. "The only countries that negotiated this deal were the United States and Russia," said press secretary Karine Jean-Pierre.

When negotiations began to secure Griner's release during the summer, the US made it clear it wanted ex-marine Paul Whelan to be included in an exchange.

But when it became clear that Whelan, who was imprisoned in 2018 on suspicion of spying, would not be included in the Russian swap, his family's hopes were dashed.

Paul Whelan told CNN he was "greatly disappointed" that more had not been done to free him, as he had committed no crime: "I don't understand why I'm still sitting here," he said.

Bout's wife, Alla, told Russian TV she had spoken to him only two days ago: "He was supposed to call me tonight. Now we'll see each other and hug each other. That's better than any phone call."

Viktor Bout sold arms to warlords and rogue governments, becoming one of the world's most wanted men.

His secretive career was brought to an end by an elaborate US sting in 2008, when he was arrested at a hotel in the Thai capital, Bangkok, to the anger of the Russian government.

He was extradited two years later and has spent the past 12 years languishing in an American jail for conspiring to support terrorists and kill Americans.

Bout's circumstances could hardly be more different from those of his opposite number in the prisoner swap.

32-year-old Brittney Griner is one of the best-known sportswomen in America. During the US basketball season, the double Olympic champion is a star centre for the Phoenix Mercury in the WNBA.

Her only reason for flying to Moscow was to play in Russia during the off-season in the US. She told her Russian trial that the cannabis oil found in her bag had been an "honest mistake".

Griner was transferred last month to a penal colony in Mordovia, a remote region 500 kilometres south of Moscow.She was held not far from where Paul Whelan is serving his 16-year jail term on spying charges.

In his statement, Biden said Russia had treated Whelan's case differently from Griner's for totally illegitimate reasons.

"While we have not yet succeeded in securing Paul's release, we have not given up; we will not give up," he vowed.

BBC