Promoted last season, Dyche's side have been among the favourites all season to go straight back down.
They have scored only one goal in seven games - the one that beat champions Manchester City - but the bottom positions are so tight that five teams are separated by only three points.
If Burnley (26 points) can beat Leicester (28), they could leap out of the bottom three, should Queens Park Rangers (26) and Hull City (28) also lose on Saturday.
Sunderland (29) are also vulnerable and even FA Cup finalists Aston Villa (32) are not safe, with five matches to play.
"The fans have been fantastic all season and I don’t use my words lightly," Dyche told the club's website.
"It's important the fans play their part. All I can ask is that they continue to give us that all the way through the rest of this season and beyond."
(Reporting By Steve Tongue; editing by Toby Davis)
Share
