The bombing of the Beth Israel synagogue was one of four attacks on British and Jewish targets this month that killed more than 60 people and injured hundreds more.
"The man was arrested as he tried to leave the country," the state-run Anatolian news agency quoted a senior police official as saying. Prosecutors have already charged several people in connection with the bombings.
Police have identified the synagogue bombers as Turkish nationals Mesut Cabuk and Gokhan Elaltuntas.
On Friday, security officials in southeast Turkey said DNA tests had revealed the identity of another of the bombers.
Habib Aktas, a Turkish national long suspected of militant Islamic leanings, was the driver of a truck laden with explosives that blew up outside the Istanbul headquarters of London-based bank HSBC on 20 November, they said.
The British consulate was also hit in on the same day.
Shadowy groups with links to the al-Qaida network have said they were responsible for all four bombings.
"It became clear that Aktas was the bomber after a blood sample taken from his father matched his"
Turkish security officer
"It became clear that Aktas was the bomber after a blood sample taken from his father matched his," a security officer said, speaking on condition of anonymity.
Police detained his father and two brothers, and they were taken to Istanbul for questioning, the official said. Dozens of others remain in police custody in connection with the bombings, and 20 people have been charged with belonging to and aiding and abetting an illegal organisation.
Six people sent to the state security court on Friday were released without charge, the Anatolian news agency said.