Venezuelan officials said they have not communicated with the NSA leaker since extending their offer to take him in.
"We would also secondly have to get in touch with the government from the Russian Federation where he is, since he is there -- and obviously not in Venezuelan territory -- to see their thoughts on it," said Venezuelan Foreign Affairs Minister Elias Jaua.
Snowden, 29, applied for protection from at least 20 countries since the leak became public and he fled from Hong Kong to Moscow.
Many were rejected, often on technical grounds, including a standard asylum requirement that the request be made within the country's border or at a border station.
He initially received an offer to stay in Russia from President Vladimir Putin, but withdrew his application over Putin's condition he "stop his work at harming our American partners."
Bolivian President Evo Morales offered protection to Snowden in response to a July 2 incident in which his plane home from Russia was diverted to Vienna, Austria after France, Portugal, Spain and Italy all closed their airspace to him allegedly because they believed Snowden was on board.
Nicaraguan leader Daniel Ortega made his offer to Snowden Friday, although he did not make it clear under what conditions his country would officially grant asylum.
"We are an open country, respectful of the right of asylum, and it's clear that if circumstances permit, we would gladly receive Snowden and give him asylum in Nicaragua," Ortega said.
For its part, Venezuela has already received and rejected an extradition request from the United States, which implored Venezuelan authorities to arrest Snowden "should he seek to travel or transit through Venezuela."
Venezuelan President Nicolas Maduro, in announcing his decision to offer Snowden asylum, took the opportunity to swat at "the empire" for what he viewed as its misdeeds beyond the broad NSA surveillance.
"He has told the truth, in the spirit of rebellion, about the US spying on the whole world," Maduro said. "Who is the guilty one? A young man… who denounces war plans, or the US government which launches bombs and arms the terrorist Syrian opposition against the people and legitimate president, Bashar al-Assad?"
"As head of state of the Bolivarian republic of Venezuela, I have decided to offer humanitarian asylum to the young Snowden… to protect this young man from persecution by the empire," he said.
It is unclear how Snowden would get from Russia to any of the countries offering him shelter, but Russia seems ready to be rid of him.
"Sanctuary for Snowden in Venezuela would be the best solution," Alexei Pushkov, the head of the Russia Duma, tweeted Saturday. "He can't live at Sheremetyevo."
Otherwise, Pushkov jested, Snowden "will have to get married to Anna Chapman."