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Nonetheless, Mr. Carney said, “it’s safe to assume, in the damage assessment [being conducted by U.S. intelligence], that any information that [Mr. Snowden] might have provided publicly, we would expect to be compromised.”

Administration officials made clear that they want Russian cooperation and are asking for it at several levels of government.

“We expect the Russian government to look at all options available to expel Mr. Snowden back to the U.S. to face justice for the crimes with which he is charged,” said White House spokeswoman Caitlin M. Hayden.

A State Department official, speaking on background, said FBI Director Robert S. Mueller III had spoken personally with his counterpart at the Russian domestic intelligence agency, the FSB, about the importance of Mr. Snowden’s return to the United States.

President Obama has in recent months pushed a “reset” policy with Russia, looking for progress on nuclear weapons reductions that he has not been able to find on Syria or other key bilateral issues.

Sen. Lindsey Graham, South Carolina Republican, warned the Russian ambassador Monday that the case was “an important test of the ‘reset’ in relations between our two countries.”

“If our two nations are to have a constructive relationship moving forward, Russian cooperation in this matter is essential,” he wrote.

Mr. Carney said authorities in Hong Kong and Beijing allowed Mr. Snowden to leave even though his passport had been revoked, and that the U.S. had filed an extradition warrant and sought his arrest.

“We are just not buying that this was a technical decision by a Hong Kong immigration official,” he said.

“The Hong Kong authorities were advised of the status of Mr. Snowden’s travel documents in plenty of time to have prohibited his travel as appropriate,” he said.

“This was a deliberate choice by the government to release a fugitive despite a valid arrest warrant, and that decision unquestionably has a negative impact on the U.S.-China relationship,” he said.

Dave Boyer and Guy Taylor contributed to this report.