Hodgson, 73, is out of contract this summer and has constantly dodged questions about his future during recent months, with his focus on a strong finish to the current campaign.
Aston Villa travel to Selhurst Park on Sunday before Arsenal visit Palace on Wednesday, with 6,500 Eagles supporters set to be allowed into the stadium for the midweek fixture which could act as a farewell on home turf for the boyhood fan.
In terms of team news, James Tomkins is closing in on a return from injury and could be involved for the Eagles.
Elsewhere James McArthur, Connor Wickham, Nathan Ferguson, Mamadou Sakho and Patrick van Aanholt all remain sidelined.
Nathanial Clyne has had a sickness bug, but could be back for Saturday.
Crystal Palace predicted XI:(4-3-3) Guaita; Ward, Kouyate, Dann, Mitchell; Eze, Milivojevic, Riedewald; Townsend, Benteke, Zaha
Injury list: McArthur, Wickham, Ferguson, Sakho, Van Aanholt
Doubts: Tomkins, Clyne