Given and his solicitor Michael Kennedy held talks with Newcastle owner Mike Ashley and managing director Derek Llambias this week.
The ambitious goalkeeper was signed from Blackburn by Kenny Dalglish in 1997, and has established himself as one of the club's most valuable assets.
Kinnear is desperate to keep Given and said: "It's out of my hands, and I am still waiting for the final outcome. But I think he wants a new challenge and that he wants to speak to Manchester City.
"I would be heartbroken to see him go, because he is one of the best goalkeepers in the world. But Man City keep banging on the door to talk to him."