The Blues reportedly joined Premier League champions Leicester in tracking the 23-year-old.
But Keane told The Guardian: "The fans have been brilliant with me here, as have the manager and players. It's great when teams like that come in but I am quite happy.
"I won't be looking at Leicester in the Champions League thinking: 'That could have been me' because I tend to think what will be, will be.
"Hopefully at some time in the future I will get to that level anyway."