An injury to Michael Duff in November handed Keane - who moved to Turf Moor initially on loan until January - a chance to hold down a regular first-team place at Burnley and the 21-year-old has impressed at the heart of defence.
Keane has been restricted to just five appearances for United, with the majority of his career spent out on loan at clubs like Blackburn Rovers and Leicester City, and Dyche is keen to keep hold of the England Under-21 international if a deal can be reached.
"We will wait and see and have a discussion with him and Manchester United," said Dyche.
"It's been ongoing since he came here."
Burnley, meanwhile, have handed fellow defender Kevin Long a new contract until June 2017.
Long made his first Premier League appearance in the 3-3 draw at Newcastle United on New Year's Day when injury forced off captain Jason Shackell, only for his own afternoon to be cruelly curtailed 20 minutes after his introduction due to a cruciate knee problem.
The 24-year-old has been ruled out for the season, but his long-term future is secure as he starts on his gruelling road to recovery.
