Snowden, who leaked details of sophisticated NSA electronic monitoring programs, left Hong Kong Sunday and traveled to Moscow before paperwork required to have him detained was completed.
Russian officials said Snowden would remain in the transit area of the Moscow airport until Monday when he is scheduled to fly to Havana.
But two passengers told the Los Angeles Times they saw a black car with red diplomatic license plates picking up some passengers from the flight's business class.
Rogers, in an appearance on NBC's "Meet the Press," said it's a serious concern that Russian President Vladimir Putin had advance knowledge of Snowden's travel plans.
"Listen, Russia is a country that wants to get back on the world stage and I don't think they really care if they do it in the best interest of good citizens around the world. This shouldn't surprise us," he said. "They have a very aggressive intelligence operation in the United States. I'm sure they would love to have a little bit coffee and a few conversations with Mr. Snowden."
"That's why this is so serious and why we need to be aggressive in making sure that people understand the difference between somebody who betrays their country and gives secrets away that will protect American lives at the expense of whatever he hopes to gain in the company of the Russians, in the company of the Chinese intelligence services, in the company of what you can only imagine is Cuban and Venezuelan intelligence services as well," Rogers said.