The centre-half was the subject of serious interest from Leicester.
Clarets boss Sean Dyche was thought to have turned down numerous offers, including one of £15m from the reigning Premier League champions.
Recent reports have suggested that Chelsea are considering a January swoop.
It is thought Antonio Conte is prepared to spend as much as £25m on the 23-year-old.
Moreover, he is seen as a long-term successor to John Terry.
However, despite Leicester’s interest over the summer, Keane harbours no grudges for a deal not going through.
“It’s flattering when teams like Leicester come in but I am really happy here,” Keane said.
“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. Leicester have done brilliantly and I will be supporting them because I was there on loan [in 2012-13] but I’m not disappointed I am not there.”
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