"An asylum request letter arrived" from Snowden, Maduro told reporters after meeting with Panamanian President Ricardo Martinelli. Snowden, he said, "will have to decide when he will fly here".
Last Friday, Maduro publicly announced his offer of "humanitarian asylum" for Snowden, accusing the US of "unleashing madness" and "persecution" after the jet carrying Bolivian President Morales was denied over flight and landing rights by several European countries who feared that Snowden might be on board the plane.
"Latin America is telling this young man that you are being persecuted by the empire; come here," said Maduro, alluding to other similar offers made to Snowden by Bolivia and Nicaragua.
Venezuela, Bolivia and Nicaragua are all governed by leftist administrations.
Maduro said that he had not spoken with Snowden personally, "but I'd like to".
When asked about possible US reprisals if Venezuela takes Snowden in, Maduro said that "the US doesn't govern the world. We're a free and sovereign country".
Snowden also made a formal asylum request of Nicaragua to that country's diplomatic delegation in Moscow.
The White House warned Monday that Snowden must not be allowed to travel to any country except the US in response to the asylum offers he has received from the three Latin American nations.
Snowden has requested refuge in 27 countries, according to WikiLeaks, after revealing massive US spying on telephone and Internet communications.
Snowden's passport was revoked by US authorities.