Home Minister Ramalinga Reddy told NDTV "We know who it is" but declined to elaborate, explaining that disclosing more information could jeopardize the case.
Ms Lakesh, 55, was shot dead as she entered her house in the IT hub of Bengaluru on September 5. A man wearing a helmet to conceal his face was recorded on security cameras as he shot her in the chest.

CCTV cameras at Gauri Lankesh's home captured a man wearing a helmet to conceal his face. (File)
Different government officials have suggested that her murder appears to share links with that of rationalist MM Kalburgi who was shot dead at his home in the Dharwad region of the state in 2015. Suspects have yet to be identified for that killing.
Sources handling the investigation into Ms Lankesh's killing said that the bullets used to kill Mr Kalburgi were sent "possibly to Scotland Yard" to identify their provenance but have not yet been returned. The details of the bullets are important to see if there is any connection between the two killings.
Ms Lankesh's killing has triggered outrage in part because some Twitter users claiming affiliation to the BJP celebrated the manner in which she was murdered. Prime Minister Narendra Modi follows some of those accounts and despite public anger, has not stopped tracking them. Celebrated southern star Prakash Raj said yesterday that the PM's silence on this issue hurts and concerns him; the actor, who was a close friend of Ms Lankesh, said that the five national awards he has won are better deserved by the PM and Uttar Pradesh Chief Minister Yogi Adityanath, whom he described as "bigger actors than me."