Mehmet Ali Agca's lawyers delivered the petition, a copy of which was obtained by Reuters, to the chief prosecutor's office in Ankara, requesting his release on January 18.
The former right-wing gangster served 19 years in an Italian prison for the assassination attempt before being pardoned at the Pope's behest in 2000.
John Paul died in 2005.
He was then extradited to Turkey to serve time in an Istanbul jail for the 1979 murder of liberal newspaper editor Abdi Ipekci and other charges dating from the 1970s.
The petition said his sentence of 40 years in prison had been based on a commuted death sentence and this was no longer valid after Turkey abolished capital punishment in 2002.
"We are seeking ... a decision to release the client on the basis that the required sentence is eight years," the petition said.
Agca was briefly freed from jail early in 2006, triggering protests in Turkey.
The Supreme Court subsequently ordered that he be returned to prison, saying it was too early for him to walk free. State prosecutors calculated January 18, 2010, as his release date.
Agca's motives for shooting Pope John Paul in the Vatican's St Peter's Square remain a mystery, but some believe he was a hitman for Soviet-era East European security services alarmed by the Polish-born pontiff's fierce opposition to communism.
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