The former Brentford player has two-and-a-half years remaining on his present contract but Hull, having reportedly valued him at £10million, are taking no chances and have opened talks over a new deal.
Turner insists he is happy to stay at the KC Stadium having made a huge impression in his side's debut Barclays Premier League campaign.
Turner said: "My agent has had a couple of meetings with (manager) Phil Brown and hopefully in the next few weeks something can be agreed.
"We are looking to extend it and hopefully something can be agreed pretty quickly.
"This is my first year in the Premier League, the club is going in the right direction and I want to be part of something here.
"First and foremost I have got to stay in the Premier League and go again next year.
"Hopefully the stadium can be developed - we have got the fanbase and the potential to be a huge club."