Palace have won just one of their last 11 matches and head to Arsenal on Saturday just five points above the drop zone.
The injury-hit Eagles are 14th in the table and have finished between 10th and 15th position every season since their return to the top flight in 2013. They went out of the FA Cup on Wednesday as Everton beat them 1-0 in a third-round replay at Goodison Park.
Asked what his side have left to play for this season, now that they are out of both cup competitions, a frosty Hodgson replied: “What do Brentford hope for, what do Fulham hope for, what do [Nottingham] Forest hope for?"
He added: “I don’t see us [finishing] in the top six. The players we have, the way we are, the organisation and group of players we’re working with, I’m certain they’ll get us clear of relegation.
“Around us we’ve got a lot of teams in the same neck of the woods as we are, but a few are a few points above us at the moment.”
Sections of Palace’s fanbase have been critical of 76-year-old Hodgson for his perceived negative selections and the team’s poor form.
“I don’t make any demands of the fans”, he said, when asked whether some supporters must be more patient.
“I can understand the other night they were upset [at me for taking Eberechi Eze off at Everton]. He gives us a better chance of winning than not, but we have to make other decisions.
“I’d have liked to say, ‘You just carry on’. But I know I’d be doing the wrong thing for him in regards to his injury and possibly the wrong thing for the club, had the injury reoccurred.
“Had the game not come so quickly maybe we’d have taken the chance, but the decision was made to play him for 60 minutes. The fans didn’t know that, but I'm pretty certain if that had been explained to the fans, they would have understood it better.
“They did a great job coming up there to support us; they were their usual vocal selves, but I can understand they must have thought, ‘This is crazy taking your best player off'. I hope I’ve explained it to them now, and I hope they can accept the explanation.”