WASHINGTON — Immediately after a Moscow judge handed down Brittney Griner’s nine-year prison sentence on Thursday, calls grew louder for President Biden to find a way to bring her home, even as critics fumed that offering to swap prisoners with Moscow rewarded Russian hostage-taking.

The result is a painful quandary for the Biden administration as it tries to maintain a hard line against President Vladimir V. Putin of Russia over the war in Ukraine.

“There’s nothing good here,” said Andrea Schneider, an expert on international conflict resolution at Cardozo School of Law. “No matter what Biden does, he’s going to be criticized — either that we’re giving too much or we’re not working hard enough.”

Kremlin officials had said talks on an exchange could not proceed before her trial was complete, but even with an official verdict and sentence, a deal may not happen anytime soon.

“I think the fact that Putin has not said yes right away means that he’s looked at the U.S. offer and said, ‘Well, that’s their first offer. I can get more than that,’” said Jared Genser, a human rights lawyer who represents Americans held by foreign governments.

The Biden administration proposed to trade Ms. Griner and Paul N. Whelan, a former Marine convicted in Moscow of espionage in 2020, for the notorious Russian arms dealer Viktor Bout, who is midway through a 25-year federal prison sentence for offering to sell arms to a Colombian rebel group that the United States then considered a terrorist organization.

Mr. Biden finds himself squeezed from two sides.

On one side are Ms. Griner’s supporters. Her wife, Cherelle Griner, has made public pleas for Mr. Biden to cut a deal with Mr. Putin as soon as possible. Those pleas have been echoed by the Rev. Al Sharpton, Democratic activist groups, television pundits, pro athletes and celebrities on social media.

But there has also been criticism from Mr. Biden’s other flank — and charges that Mr. Biden has been bending to extortion by Mr. Putin, a man he has called a war criminal.

“This is why dictatorships — like Venezuela, Iran, China, Russia — take Americans hostage, because they know they’ll get something for it,” Rep. Mike Waltz, Republican of Florida, told Newsmax last week. “They know eventually some administration will pay. And this just puts a target on the back of every American out there.”

Mike Pompeo, the former secretary of state, echoed the criticism in a Fox News interview last week, saying that to free Mr. Bout would “likely lead to more” Americans being arrested abroad.

And former President Donald J. Trump, who is likely to run again in 2024, slammed the proposed deal in crude terms. He said Mr. Bout was “absolutely one of the worst in the world, and he’s going to be given his freedom because a potentially spoiled person goes into Russia loaded up with drugs.” (Russian officials who detained Ms. Griner at a Moscow-area airport in mid-February found less than one gram of cannabis vape oil in her bags.)