The assailant stormed the glamorous Reina nightclub on the Bosphorus early Sunday morning, spraying 120 bullets at terrified partygoers celebrating the start of 2017.
Of the 39 dead, 27 were foreigners including citizens from Lebanon, Saudi Arabia, Israel, Jordan, Iraq, Tunisia and Morocco.
At least 36 people have now been detained in the probe, but the gunman himself remains on the run after slipping into the night following the attack.
Turkey again extends emergency rule by further three months
"The identity of the person responsible for the attack has been established," Foreign Minister Mevlut Cavusoglu said during an interview with state-run Anadolu news agency, without giving any name.
The attack was claimed by the Islamic State (IS) group, with reports suggesting the authorities suspect the gunman may be from either Kyrgyzstan or Uzbekistan, both ex-Soviet states.
"Efforts to capture him continue," said Cavusoglu, adding that the house the suspect lived in "has been searched" and that the attack he mounted had been "professionally" planned.
IS took responsibility for the massacre in a statement on Monday, marking the first time it has issued a clear and undisputed claim for a major attack inside Turkey.
COMMENTS
Comments are moderated and generally will be posted if they are on-topic and not abusive.
For more information, please see our Comments FAQ