Dyche, who took charge of Burnley back in 2012 after one season at Watford, has received rave reviews for his managerial performance at Turf Moor, keeping the Clarets in the Premier League.
The English tactician has done superbly to get the best out of his squad and there are said to be a few teams that are considering making a move for the 45-year-old this summer.
Southampton are currently searching for a new boss following the sacking of Claude Puel, while Palace's search for Sam Allardyce's replacement has been going on for nearly a month now.
Numerous names have been linked with the position at Selhurst Park, including former Inter boss Frank de Boer and Mauricio Pellegrino, who was last seen in charge of Alaves.
However, Dyche is believed to be keen on discussing the possibility of taking over the Eagles although there is yet to be any contact made between the manager and club officials.