The future of their boss Roy Hodgson is unclear, even though he recently signed a contract until 2021.
Hodgson is 72 and it is unclear how long he plans to continue in management, especially amid the COVID-19 outbreak.
The Daily Mirror claims Dyche is seen as the ideal successor to Hodgson.
Palace do hope that Hodgson sticks around for another season or two.
But they are already making plans for life after the former England boss.
For his part, Dyche could be at the end of the line with Burnley, given their limited finances and inability to push up the Premier League table.
He admitted on the Eamonn and the Gaffers podcast recently that he could leave Burnley for the right offer.