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.

“We call on President Biden and the United States government to redouble their efforts to do whatever is necessary and possible,” the Rev. Al Sharpton said in a statement.

U.S. officials and analysts had been resigned to a guilty verdict for Ms. Griner, a basketball star who plays for a Russian team during the W.N.B.A. off-season. But the cold reality of her sentence on a drug charge was a shock and renewed calls for Mr. Biden to secure her release — even as critics fumed that offering to swap prisoners with Moscow rewards 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 his 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 that any potential deal could not proceed before her trial was complete, creating a glimmer of hope that the verdict might open the door for an exchange. But analysts called that unlikely anytime soon.

“I don’t think this is going to get resolved quickly,” said Jared Genser, a human rights lawyer who represents Americans held by foreign governments. “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.’”

That U.S. offer, first presented to Russia in June, sought the release of Ms. Griner and Paul N. Whelan, a former Marine arrested in Moscow and convicted of espionage in 2020.

The Biden administration proposed to trade the two Americans 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.

The proposal has already reshaped U.S. diplomacy toward Russia, which had been frozen at senior levels since Mr. Putin’s Feb. 24 invasion of Ukraine. A phone call about the matter on July 29 between Secretary of State Antony J. Blinken and his Russian counterpart, Sergey V. Lavrov, was their first conversation since the war began. But it appeared to leave the Kremlin unmoved. The White House says Russia has made an unspecified “bad faith” counteroffer that the United States is not taking seriously.

On Friday, Mr. Lavrov told reporters that the two nations would continue discussing the issue through established channels. He repeated the Kremlin’s insistence that the United States not discuss the negotiations in public, though Russian media outlets began linking Mr. Bout’s case to Ms. Griner’s early this summer.

But the pressure is lopsided. While Mr. Putin has long sought Mr. Bout’s release, perhaps out of loyalty to a man with deep ties to Russia’s security state, the arms dealer’s continued imprisonment costs Mr. Putin little. Time, in other words, is in Mr. Putin’s favor.

Mr. Biden, on the other hand, 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 Mr. Sharpton, Democratic activist groups, television pundits, pro athletes and celebrities on social media. (Mr. Sharpton on Thursday also called for the release of Mr. Whelan.)

“How could she feel like America has her back?” the N.B.A. superstar LeBron James said in mid-July. “I would be feeling like, ‘Do I even want to go back to America?’”

That was before Mr. Biden’s proposal to free Mr. Bout became public. Officials said they disclosed the offer, which was confirmed last week by a person briefed on the talks, to increase pressure on Russia. But the revelation may have also reflected a desire to show Ms. Griner’s backers that Mr. Biden was not sitting on his hands.

“We believe it’s important for the American people to know how hard President Biden is working to get Brittney Griner and Paul Whelan home,” John F. Kirby, a White House national security spokesman, said at the time. “We think it’s important for their families to know how hard we’re working on this.”

After Ms. Griner was sentenced on Thursday, Mr. Biden renewed his commitment to “pursue every possible avenue to bring Brittney and Paul Whelan home safely as soon as possible.”

The White House would not say how Mr. Biden might achieve that goal, however. “I don’t think it would be helpful to Brittany or to Paul for us to talk more publicly about where we are in the talks and what the president might or might not be willing to do,” Mr. Kirby said.

But almost any additional offers would be sure to amplify criticism from Mr. Biden’s other flank — and charges that Mr. Biden was 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 when in office prided himself on freeing detained Americans abroad, slammed the proposed deal in crude terms.

Mr. Bout, he said, 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.)

Mr. Genser, the lawyer for other detained Americans, noted that Mr. Biden has an option beyond raising his offer. He could seek new ways to make Mr. Putin suffer.

“You need to dramatically elevate the cost to Vladimir Putin of keeping them detained,” Mr. Genser said. “It’s not only about giving Putin what he wants. It’s about simultaneously raising the pain for him.”

That is no easy task, however. Biden administration officials have spent months trying to devise ways to incur enough pain on Mr. Putin to make him cease his invasion of Ukraine. Like the freedom of Ms. Griner and Mr. Whelan, that goal, too, remains elusive.