Hodgson has confirmed Wilfried Zaha does not need surgery on his knee injury and Palace hope the forward will return before the end of next month.
But Palace fear Julian Speroni will be out for two months with his knee injury and Hodgson has targeted a back-up for Wayne Hennessey over an extra attacking option.
Palace have space left in their 25-man Premier League squad to register one player and they can sign a free agent outside of the transfer window.
Hennessey is the only fit senior goalkeeper at Palace after Speroni injured his knee in training and 20-year-old Dion-Curtis Henry will be on the bench at Everton tomorrow.
Palace have been left to rue their failure to sign a goalkeeper in January and the Eagles are now considering what goalkeepers are available.
Hodgson made no secret of his desire to sign a goalkeeper last month but Palace missed out on Vicente Guaita from Getafe and Frederik Ronnow from Brondby on deadline day.
Palace have reached an agreement to sign Guaita in the summer but Hodgson is concerned about the lack of options for the relegation run-in, especially given Hennessey has been troubled by back spasms during his career.
Henry has been a regular for the Under-23s this season but has not made a senior appearance since joining from Peterborough United last August.
The squad has been ripped apart by injuries and Zaha is the latest out of Saturday’s trip to Everton.
Hodgson will face Sam Allardyce for the first time since the former Palace boss criticised him in the undercover newspaper investigation that cost him the England job.
Allardyce, now Everton boss, was asked in 2016 why Hodgson would not be a good public speaker and replied: “He’d send them all to sleep, Roy. Woy. He has not got the personality for it.”
Allardyce also said Hodgson was “too indecisive” during England’s shock Euro 2016 defeat by Iceland.
Hodgson has revealed Allardyce never apologised and said: “My relationship with him before I always thought was good. Now I would expect it to be less good.
Asked if he would like Allardyce to apologise, Hodgson said: “No, [it is] far too late. It has gone. It belongs to the distant past.”