It's believed Palace chairman and part-owner Steve Parish is stepping up plans for a change after a horror run of seven straight defeats coupled with no goals in their last three games, says the Mirror.
Hodgson, 72, is under the microscope with Palace facing Tottenham at Selhurst Park on Sunday in their final Premier League match of the campaign.
Palace are safe – but it's understood Parish has been less than impressed with results even though five of the seven defeats were against teams in the top five.
Parish has remained loyal to Hodgson who signed a new one-year deal in March.
But there's a boardroom belief it's time for a managerial change – with Dyche firmly in the frame.