On Monday, Palace announced that Frank de Boer had been removed from his position in the dugout at Selhurst Park after just four games, and Hodgson is expected to be named as the Dutchman's replacement.
However, Barton has been critical of Hodgson, claiming that former England boss does not have the characteristics to move the Eagles up the league table.
The 35-year-old told talkSPORT: "He's just got no charisma, no personality at all! Did you see him at the back end of that England campaign? He just looked like he needed to be taken out into the back garden and have someone pull the trigger on him.
"A lot of people say he's a great fella, but I just don't think [he's the right fit for the Palace job]. I didn't see the Frank de Boer appointment, I thought it was a bit left field and I think if you're them you want somebody who lifts the place up."
Palace are currently three points adrift of safety in the top-flight standings.