Police released pictures of the suspect who went on the rampage at the plush Reina nightclub on New Year's night, spraying about 120 bullets at terrified partygoers before slipping away into the night.
So far, 16 people are being held over the attack, including two foreigners detained by Turkish police at Istanbul's main airport.
But the killer remains on the run.
There was frenzied speculation surrounding a 28-year-old Kyrgyz man with a strong facial resemblance to the attacker but he was allowed to fly back home by the Turkish authorities and later released after questioning in Kyrgyzstan.
More From This Section
The Islamic State (IS) group yesterday claimed the massacre, the first time it has clearly stated being behind a major attack in Turkey.
The suspect -- who has not been named but reportedly may be from Kyrgyzstan or Uzbekistan -- was staying in a rented flat in Konya before moving to Istanbul to carry out the attack, press reports said.
The Dogan news agency said those detained included a woman suspected of being his wife with whom he had stayed in Konya along with two children.
Reports said police have made progress in the investigation after speaking to the taxi driver who drove the attacker to the club and tracing calls he had made on the driver's mobile phone.
The Hurriyet daily said the attacker showed signs of being well trained in the use of arms and had fought in Syria for IS jihadists.
Hurriyet's well-connected columnist Abdulkadir Selvi said the suspect had been trained in street fighting in residential areas in Syria and used these techniques in the attack, shooting from the hip rather than as a sniper.
"This specially-trained terrorist has still not been detained and is still wandering dangerously amongst us," Selvi wrote.
He said an IS strike was also planned in Ankara on New Year's eve but that it had been prevented after eight IS suspects were arrested in the capital.
Together with a unexpected jump in inflation, anxiety over the attack pushed the Turkish lira to a new historic low of 3.6 to the US dollar.
Near the entrance to the nightclub on the shores of the Bosphorus, an impromptu shrine was set up with pictures of the dead where well-wishers have been piling up flowers.