The lawmaker, Alexei Pushkov, later tweeted that he had seen the news on state-run Russian television channel Rossiya-24, but a representative of Rossiya-24 said it had been referring to Pushkov's initial tweet.
Pushkov, the pro-Kremlin chairman of the international affairs committee in the lower parliament house, could not immediately be reached for comment.
A Russian Foreign Ministry official declined immediate comment and the Venezuelan embassy in Moscow could not immediately be reached.
Snowden, who is wanted in the United States on espionage charges after revealing details of secret surveillance programs, is believed to be holed up in the transit area at a Moscow airport where he arrived on June 23 from Hong Kong.
He has appealed to about 20 countries for asylum and Venezuelan President Nicolas Maduro said on Friday that he had decided to offer the 30-year-old American fugitive asylum. Nicaragua and Bolivia also have said they would take him in.
Russian President Vladimir Putin has said Snowden should choose a final destination and go there as soon as possible, but it is unclear how he would get to any of the Latin American countries that have offered him asylum.
There are no direct flights from Moscow to Venezuela, Nicaragua or Bolivia, and U.S. authorities have urged nations around the world not to give Snowden refuge.
(Reporting by Gabriela Baczynska; Writing by Steve Gutterman; Editing by Michael Roddy)