Snowden's options aren't numerous, said Assange's lawyer, Michael Ratner.
"You have to have a country that's going to stand up to the United States," Ratner said. "You're not talking about a huge range of countries here."
It also wasn't clear Snowden was finished disclosing highly classified information.
Snowden has perhaps more than 200 sensitive documents, Sen. Dianne Feinstein, chairwoman of the Senate Intelligence Committee, said on CBS' "Face the Nation."
Associated Press White House Correspondent Julie Pace and Associated Press writers Matthew Lee and Frederic J. Frommer in Washington, Lynn Berry in Moscow, Kevin Chan in Hong Kong and Sylvia Hui in London contributed to this report.