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 his country hasn't 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 U.S. 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 byVenezuela'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 U.S., 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 doesn't 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 U.S. in 2010. The 31-year-old Chapman proposed to Snowden, who just turned 30, on Twitter last week.
The presidents of Bolivia and Nicaragua also said over the weekend that Snowden was welcome in their countries. Bolivia's foreign minister, David Choquehuanca, said Sunday on state television that his country hasn't yet received a formal petition for asylum from Snowden. Nicaraguan President Daniel Ortega said his country's embassy in Moscow has received Snowden's application and is studying the request.
Snowden has applied for asylum in more than two dozen countries, including Bolivia, Nicaragua andVenezuela, according to WikiLeaks, the secret-spilling website that has been advising him.
U.S. Sen. Robert Menendez, chairman of the Senate Foreign Relations Committee, said he wasn't surprised that those three Latin American nations were offering asylum.
"They like sticking it to the United States," Menendez told NBC's "Meet the Press."
He also mentioned re-examining U.S. trade policies and foreign aid to any country that might take inSnowden.
Brazil's foreign minister said his government is worried by a newspaper report the U.S. has collected data on billions of telephone and email conversations in his country and promised an effort for international protection of Internet privacy.
"The Brazilian government has asked for clarifications" through the U.S. Embassy in Brazil and Brazil's embassy in Washington, Foreign Minister Antonio Patriota said.
The spokesman for the U.S. Embassy in Brazil's capital, Dean Chaves, said diplomats there wouldn't have any comment.