Alexei Pushkov, who heads the international affairs committee in Russia's parliament, posted a message on Twitter saying: "Venezuela is waiting for an answer from Snowden. This, perhaps, is his last chance to receive political asylum."
Venezuelan Foreign Minister Elias Jaua said Saturday that his country has not yet been in contact with Snowden, who Russian officials say has been stuck in the transit area of Moscow's Sheremetyevo Airport since arriving on a flight from Hong Kong two weeks ago. He has been unable to travel further because the United States annulled his passport.
Jaua said he expects to consult with Russian officials on Monday about Snowden's situation.
Pushkov's comments appeared to indicate that the Kremlin is now anxious to be rid of the former National Security Agency systems analyst, whom the U.S. wants returned to face espionage charges.
There has been no response from the Kremlin or Russian Foreign Ministry to the asylum offer made by Venezuela's President Nicolas Maduro in the early hours of Saturday, Moscow time.
For Snowden to leave for South America, he would need for Venezuela to issue him travel documents and he would need to find a way to get there. The only direct commercial flight from Moscow goes to Havana, Cuba, and Snowden had booked a seat on this flight the day after arriving from Hong Kong, but failed to show up.
The Moscow-Havana flight goes over Europe and the United States, which could cause complications. Some European countries refused to allow Bolivian President Evo Morales to fly through their airspace on his way home from Moscow last week because of suspicions that Snowden was onboard his plane.
Pushkov joked that if Snowden does not find shelter in Venezuela, "he will have to stay and marry Anna Chapman," the redheaded Russian spy who was among 10 sleeper agents deported from the United States in 2010. The 31-year-old Chapman proposed to Snowden, who just turned 30, on Twitter last week.
Associated Press writer Fabiola Sanchez in Caracas, Venezuela, contributed to this report.