The Eagles opted to bring in the French coach over the summer following Oliver Glasner's exit on the back of winning the UEFA Conference League.
However, Sage has struggled so far, with just two wins from eight Premier League games, and one victory from three played in the Europa League.
Dyche has been out of management since February, after being dismissed by Forest, but Palace would need to work hard to convince him over a return to the dugout.
Several key figures at Selhurst Park are open to making a change now - as per Telegraph Sport - to prepare the club for a possible fight against relegation in 2027.
Dyche had an excellent record in relegation battles during his decade at Burnley, after he got the club promoted to the Premier League twice, and kept them in the top flight for six consecutive seasons before his exit in 2022.