The Norwegian branch of the PEN Club has invited Snowden, who has been living in exile in Russia since 2013 after revealing widespread US foreign surveillance, to collect the Ossietzky prize for freedom of expression in November.
PEN said a law firm had filed a petition with Oslo City Court "in order to allow Snowden to travel to Norway without fear of extradition to the US, where he faces decades of imprisonment under the Espionage Act".
"We will do our utmost to ensure that Snowden may receive the prize in person," it said in a statement.
The 32-year-old American was charged by US authorities with espionage and the theft of state secrets after revealing the extent of surveillance programmes run by the National Security Agency.
Norway's justice ministry said it had not yet received the lawsuit.
"We never comment on individual extradition cases, let alone when they have not been examined in court," said ministry spokesman Gunnar Johansen.
Considered a whistleblower by some and a traitor by others, Snowden won a similar Norwegian award last year.
He was unable to collect it in person after he failed to receive assurances that he would not be arrested and extradited to the United States.
Snowden has also been nominated again for this year's Nobel Peace Prize, which will be announced in October.