What is clear is that the longer the situation remains unresolved, the more it could fray U.S.-Russian ties.
The former Cold War-foes are already at odds over human rights and Putin's treatment of opponents and have squared off over the Syria conflict in the U.N. Security Council.
Hagel reiterated criticism of China over Snowden's departure from Hong Kong. "We're very disappointed in the Chinese government in how they've handled this. And it could have been handled a different way," he said.
The United States has no extradition treaty with Moscow, but says there is a clear legal basis for Snowden to be handed over. White House spokesman Jay Carney said on Wednesday that U.S. and Russian officials were "having conversations" on the issue, but declined to give details.
Carney told reporters Washington could understand that Snowden's decision to go to Moscow "creates issues the Russian government has to consider."
"We also believe that when it comes to Mr. Snowden, well, we agree with President Putin that we don't want the situation to harm our relations," Carney said while travelling with Obama to Africa.
Russian Foreign Minister Sergei Lavrov repeated Putin's view that Snowden should choose a destination and fly out as soon as possible, state-run Itar-Tass news agency reported.
Putin, a former KGB officer, may feel little sympathy for someone who has broken the secrecy code. He has suggested the surveillance methods revealed by Snowden were justified in fighting terror, if carried out lawfully.
Russia appears in no hurry
But Snowden could be a useful propaganda tool for Moscow, which accuses the United States of violating rights and freedoms it vocally urges other countries, including Russia, to protect.