US was using a wide range of spying methods including bugging, the Guardian daily in London said quoting the leaked report.
"One document lists 38 embassies and missions, describing them as 'targets'. "It details an extraordinary range of spying methods used against each target, from bugs implanted in electronic communications gear to taps into cables to the collection of transmissions with specialised antennae," the Guardian said.
It added, "along with traditional ideological adversaries and sensitive Middle Eastern countries, the list of targets includes the EU missions and the French, Italian and Greek embassies, as well as a number of other American allies, including Japan, Mexico, South Korea, India and Turkey.
"The list in the September 2010 document does not mention the UK, Germany or other western European states".
It said that one of the bugging methods mentioned is codenamed Dropmire, which, according to a 2007 document, is "implanted on the Cryptofax at the EU embassy, DC" an apparent reference to a bug placed in a commercially available encrypted fax machine used at the mission.
The NSA documents note the machine is used to send cables back to foreign affairs ministries in European capitals. Snowden, 30, had blown the lid off National Security Agency's secret spy programme and is charged with violating American espionage laws.
He is currently in Moscow airport after fleeing from Hong Kong. Snowden worked as a Hawaii-based computer network administrator for Booz Allen Hamilton before he fled to Hong Kong last month with laptops full of confidential information.
The documents revealed the existence of programmes that collect records of domestic telephone calls in the US and monitor the internet activity of overseas residents.
The disclosures shook the US intelligence community and raised questions about whether the NSA is eroding American civil liberties.
Snowden is charged with theft of government property, unauthorised communication of national defence information and willful communication of classified communications intelligence. Snowden has applied for asylum in Ecuador.
The US has revoked his passport.