The Eagles are currently 15th in the Premier League table and are five points above the relegation zone.
Hodgson, 76, took charge of his 200th match as Palace manager earlier this week but his side were beaten 3-1 at Selhurst Park by Chelsea.
Palace face Everton at Goodison Park in the Premier League on Monday.
Hodgson is due to speak to the media on Thursday, but it is not certain whether he will take charge of the match on Merseyside.
More to follow.