Although Celik said he killed Peshkov, he refused to give further information regarding the incident, the source said.
The man also added that he doesn't regret killing the Russian pilot and given an opportunity he'd do the same thing again, Dogan News Agency said.
However, another Turkish newspaper Haberturk reported that Celik, in fact, refused to confirm the killing of Peshkov during an interrogation process.
Earlier it was reported that Celik was arrested in the Turkish city of Izmir upon his arrival from Syria, where he was fighting against the country's government forces alongside Turkmen militants.
According to the Hurriyet Daily News, Celik and 13 others were having dinner at a restaurant in Izmir's Hatay district when they were arrested by local police on March 30.
It was already after his arrest that it became clear that Celik was a suspect in the killing of the Su-24 pilot.
The Russian Su-24 bomber was shot down by a Turkish F-16 fighter jet over Syria on November 24, 2015 and fell four kilometers (2.5 miles) from the Turkish border. The crew of the plane ejected and one of its pilots, Peshkov, was killed by fire from the ground, while the second pilot survived and was later rescued.