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“The Chinese have emphasized the importance of building mutual trust,” he said. “We think that they have dealt that effort a serious setback. If we cannot count on them to honor their legal extradition obligations, then there is a problem.”

Secretary of State John F. Kerry said Tuesday that Mr. Putin is correct about the lack of an extradition treaty, but it is common international custom for nations to turn over foreign fugitives to their home countries to face charges there.

“I would simply appeal for calm and reasonableness,” Mr. Kerry said during a visit to Saudi Arabia. “We would hope that Russia would not side with someone who is a fugitive from justice.”

Sen. John McCain, Arizona Republican and a frequent critic of Mr. Obama’s foreign policy, said the Snowden case was the latest in a series of incidents that show growing U.S. weakness abroad.

“For nearly five years now, we have sent a signal to the world that we’re leading from behind, that we are impotent, that we don’t act when we say that we’re going to,” Mr. McCain said Tuesday on CNBC.

He called Mr. Putin “an old KGB colonel apparatchik” who “continues to stick his thumb in our eye.”

Mr. Snowden’s visits to China and Russia also raise the risk that he might divulge other classified secrets about U.S. surveillance programs to those governments. Mr. Putin said the Russian government had not debriefed the former CIA employee.

Russia’s special security services “are not engaged with him and will not be engaged,” Mr. Putin said.

Andrei Soldatov, a Moscow-based specialist on Russia’s special services who runs the website, said he doubted that Mr. Snowden was being asked to cooperate. “The Snowden case is much more important for politicians than it is for foreign intelligence services,” he told the Financial Times.

Mr. Aron said the U.S. has bigger problems with Russia, such as its support for Syria and Iran, and that the Snowden case isn’t likely to cause lasting harm compared with those international hot spots.

He also said the Obama administration doesn’t want to jeopardize the president’s goal of nuclear arms reduction, which he outlined in a speech in Germany last week.

“It appears that this administration’s almost sole focus with regard to Russia is the next nuclear-arms reduction agreement,” Mr. Aron said. “This is a paramount, overriding objective of this administration, and it would overlook a great deal to get to that point.”