The current terms of the 73-year-old's deal will expire in the summer and he will depart Selhurst Park after the final match of the campaign against Liverpool on Sunday, saying he will not manage in the top flight again.
Hodgson, who started out as a manager in 1976 with Halmstad in Sweden, has overseen top-flight stability since he was summoned by the Eagles in September 2017 and has helped Palace become an established member of the Premier League.
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Wilder, who left United earlier this year, is one name in the bookies’ odds lists, also featuring the likes of Frank Lampard, Sean Dyche and Barnsley boss Valerian Ismael.
Hodgson told Palace’s official website: "After more than 45 years of coaching I have decided that the time is right for me to step away from the rigours of top-flight Premier League football, so our final two matches will be my last ones as manager of Crystal Palace.
"It's been a particularly rewarding period of my football life and career to have been able to spend these last four seasons with Palace.
“I feel now that at the end of another successful season, in which we have secured our Premier League status, the moment is right for me to step down from my responsibilities of being a full-time manager.
"I have had so much support from my wife and family throughout my career and I have been contemplating this decision for some time, and now I believe the time is right to consider them and to see what the future holds for me."
Palace chairman Steve Parish added: "It has been an absolute privilege and pleasure to work alongside Roy, who is both a magnificent human and an outstanding football manager.
“I know how much it has meant to Roy managing the club he supported as a child, adding to his distinguished and unparalleled career in football management.”