The Clarets are worried about Dyche being tempted to leave when the current season concludes.
Dyche reportedly is attracting interest from the likes of Crystal Palace, and may leave for a club with a higher transfer budget.
Wilder was recently sacked by the Blades as they are facing relegation from the Premier League.
Everything depends on what Palace choose to do with their boss Roy Hodgson, who is out of contract in the summer.
If they choose to move on from Hodgson, Dyche could go to the London club, and that would open the door for Wilder at Burnley.