Lawyer Mustafa Demirag on Tuesday said he was seeking permission under a Turkish law that allows 72 hours leave to prisoners who have served a quarter of their term and have shown "good behaviour".
He submitted the letter to a local prosecutors' office.
Demirag, however, acknowledged earlier that there was little chance that Agca, who shot the Pope in a 1981 assassination attempt and earlier killed a top Turkish newspaper editor, would be allowed to leave prison.
The Pope met Agca in an Italian prison in 1983 and forgave him for the shooting.
Agca was extradited to Turkey in 2000 after almost 20 years behind bars in Italy. He is currently serving a 17-year prison sentence in Istanbul for earlier crimes in Turkey.
In a separate handwritten letter to the prosecutor, dated 4 April, Agca said he was seeking permission to fly to Rome in a private jet, accompanied by Interpol police officers.
"I expect the Turkish government will understand the vital importance of the issue and will urgently give its consent"
Mehmet Ali Agca
"I believe that the Vatican and Italian governments will agree to my request," the letter said.
"I expect the Turkish government will understand the vital importance of the issue and will urgently give its consent," it said.
Demirag said he was hoping to get a reply later on Tuesday.
Demirag met with Agca in his cell in Istanbul's Kartal prison on Monday and quoted Agca as saying: "I must be there. I must attend the funeral. If I can't go, then someone from my family should go."