The meeting is expected to include U.S. Secretary of State John Kerry, Secretary of Defense Chuck Hagel and their Russian counterparts. Agenda topics include missile defense, nuclear arms reductions and the crisis in Syria, the official said.
Separately Pentagon spokesman George Little told reporters, "We obviously want to maintain our relationship with the Russian military."
He said he had no comment about any upcoming meetings with the Russians.
Another U.S. official who also declined to be named said that the Russian government "has been signaling ... for some time" that it planned to grant Snowden temporary asylum. "I don't think it's a shock," the official told CNN's Barbara Starr.
While the Russians have signaled their intentions publicly, the U.S. also learned of the country's plans in private conversations between senior U.S. and Russian officials over the last several weeks, the official said.
Senior White House, Justice Department, State Department and CIA officials had been speaking with their counterparts in Russia since the crisis developed. The official did not, however, say that the U.S. had been formally notified of the decision in advance.
While the Obama administration has to make a decision about what to do next in its relations with Moscow, the U.S. national security agencies are hoping to continue cooperation with Russia on counterterrorism matters. The official said the April Boston terror attack "reinforced the need for that."
The U.S. and Russia also are already cooperating on security for the 2014 Sochi Olympic Games, which the official noted are already being threatened by Chechen terrorists.
"Russia's action today is a disgrace and a deliberate effort to embarrass" the U.S., Sen. John McCain said in a statement.
"It is a slap in the face of all Americans," McCain said, calling for the U.S. to fundamentally rethink" its relations with Russian President Putin.
Harsh reactions from lawmakers continued to pour in.
"Russia has stabbed us in the back, and each day that Mr. Snowden is allowed to roam free is another twist of the knife," said New York Sen. Charles E. Schumer in a statement. "Others who have practiced civil disobedience in the past have stood up and faced the charges because they strongly believed in what they were doing.
"Mr. Snowden is a coward who has chosen to run," Schumer said. "Given Russia's decision today, the President should recommend moving the G-20 summit."
Snowden is a "fugitive who belongs in a United States courtroom, not a free man deserving of asylum in Russia," Sen. Robert Menendez, chairman of the Senate Foreign Relations Committee, said in a statement.
Russia's action is "a setback to U.S.-Russia relations," said Menendez, D-New Jersey. Snowden "will potentially do great damage to U.S. national security interests," and the leaked information "could aid terrorists," he said.
A CNN journalist saw the Senate Armed Services Chairman Carl Levin in the hallway at the Capitol building and asked him what he thought. "I'm sorry they offered it (asylum) to him," Levin answered.
CNN asked how the move might affect relations between U.S. and Russia.
"It doesn't help them," he answered. "I hope it doesn't set them back too far because they already have plenty of challenges."
Republican Sen. Lindsey Graham, in a statement, said that "Americans in Washington should consider this a game changer in our relationship with Russia."
Russia "could not be more provocative," and Thursday's developments in Snowden's case show that Putin has a "clear lack of respect for President Obama," Lindsey said.
He called on Congress and the Obama administration to make it clear to Russia that the U.S. will react in a "firm" way.
Russia media quotes Snowden's father
Snowden's father Lon Snowden, who has adamantly supported his son, appeared on state-run Russia 24 Thursday. He's grateful to Russia and the country's decision to grant asylum is worth respect, he said.
He loves his son, he added, and is looking forward to getting a visa to visit Moscow.
In fact, the father plans to see his son there soon, said Bruce Fein, the father's attorney.