An image of a Kyrgyz passport purportedly belonging to the suspect circulated on social media.
But Kyrgyz authorities said they had questioned the man and that he had played no part in the attack.
Turkish media have run images of a suspect, saying they were given out by the police.
But the police have given no official details of the search for the suspect.
The passport image circulated on social media shows a similar-looking man to the suspect, but the head of the Kyrgyz security services told the BBC that the passport holder was not involved.
Rakhat Sulaimanov said the man had been questioned by both their agents and Turkish security and that he had nothing to do with the attack.
The Kyrgyz authorities say they were not contacted by their Turkish counterparts about the possible connection. Tairbek Sarpashev from the Kyrgyz state registration service, said they only started to investigate the man in question after seeing the media reports.
Turkish media have said the man police are investigating is from Central Asia and had rented a flat in the central city of Konya with a woman believed to be his wife and two children, who have been detained.
As more details emerge about the Istanbul nightclub gunman, it looks increasingly likely that - unlike most of the attacks in France and Germany last summer - the murderer had at least some rudimentary military training.
Like the IS gunmen who attacked Paris in November 2015 and the jihadists who attacked Mumbai in 2008, he arrived on the scene with several full magazines for his assault rifle, each containing 30 rounds.
Reports from the scene say he fired more than 100 rounds, shooting at the wounded and setting off explosive devices as a distraction as he reloaded. After nearly six years of civil war in Syria, and a still-active insurgency in Iraq, governments in both Europe and the Middle East are deeply concerned that fanatical jihadists who have spent time in those arenas will try to import their lethal skills into other countries.
Some reports suggest police have gathered details of the man from calls he made on the mobile phone of the taxi driver who took him to the nightclub.
However, nothing has been officially confirmed as Turkish authorities have given no details.
The only official statements came from Deputy Prime Minister Numan Kurtulmus, who said on Monday that authorities had fingerprints and a basic description of the suspect.
Twelve people were arrested on Monday and another two foreigners were detained at Istanbul's Ataturk airport on Tuesday.
Late on Tuesday, Turkey's parliament extended a state of emergency imposed following a failed military coup in July.  It will now be in force until at least mid-April.
Mr Kurtulmus said the extension was necessary as followers of the US-based Muslim cleric Fethullah Gulen - whom the government blames for instigating the coup - were still active in public institutions.
He added: "Regardless of which terrorist organisation they are (...) and regardless of their motivation, we are determined to fight all terrorist organisations in 2017 and we will fight till the end."
The Islamic State (IS) group has said one of its militants carried out the Istanbul attack and a huge manhunt is now underway to find him, with officials at airports and border crossings on high alert.
The gunman arrived at the club by taxi early on Sunday, before rushing through the entrance with a long-barrelled gun he had taken from the boot of the car.
The attacker fired randomly at people. About two-thirds of those killed were foreign, according to local media.
The victims
20 August 2016: Bomb attack on wedding party in Gaziantep kills at least 30 people, in an attack claimed by IS
28 June 2016: A gun and bomb attack on Ataturk airport in Istanbul kills 41 people, blamed on IS militants
19 March 2016: Suicide bomb kills four people in shopping street in Istanbul. IS blamed
12 January 2016: 12 Germans killed by IS bomber in tourist area of Istanbul
10 October 2015: More than 100 killed at peace rally outside railway station in Ankara; blamed on IS
20 July 2015: 34 people killed in bombing in Suruc, near Syrian border; IS blamed
Kyrgyzstan has ruled out a suspect named in media reports in connection with the new year attack on an Istanbul nightclub that killed 39 people.