Greenwald, an American who was among the first to release details of Washington's vast electronic surveillance program, gave no details of the content of the files as he testified before the Brazilian Senate's foreign relations committee.
"I did not do an exact count, but he gave me 15,000, 20,000 documents. Very, very complete and very long," Greenwald said, responding to questions from lawmakers.
"The stories we have published are a small portion. There will certainly be more revelations on the espionage activities of the US government and allied governments (...) on how they have penetrated the communications systems of Brazil and Latin America," he said.
Last month, US Vice President Joe Biden called Brazilian President Dilma Rousseff to provide an explanation following press reports of US electronic surveillance in Brazil based on leaks from Snowden, a former US National Security Agency contractor.
Snowden is now a fugitive from US justice and is currently living at an unknown location in Russia after Moscow granted him temporary asylum for a year.