However, former Liverpool defender Carragher couldn't resist the opportunity to poke fun at ex-Manchester United star Neville.
"Why appoint a manager like that if you knew him losing his first few would mean you would sack him?" he tweeted.
The tributes to de Boer's reign with Crystal Palace are pouring in as we speak. The Eagles have not only lost all four matches, but are the only team in the division to not score a goal this term.
Hodgson has been out of work after his abysmal performance for England at the Euro 2016 Nations Cup.
Palace chairman Steve Parish was at Turf Moor on Sunday, and said afterwards: "It's a awful start but we have to stick together".
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But that deal fell through, prompting Wenger to call off the Sanchez sale and put his player in a hard position. The game started out fairly evenly, with both teams not having much of a clear-cut goal-scoring opportunity.
The Eagles are still searching for their first goal and point of the season under new manager Frank de Boer, following three straight defeats.
Writing on Instagram, he said: 'Very disappointed about the decision but never the less I want to thank the players, staff and the fans for their support.
"For as many managers that have a great start there are probably as many that don't.
Believe me, my time is very well filled indeed".
"I wouldn't be comfortable at this moment in time taking a position as a football manager".
Meanwhile, Sam Allardyce has ruled himself out of the running for a return to Crystal Palace, who revealed on Monday Night Football that he spoke to chairman Steve Parish on Monday afternoon. We had a much weaker squad when we finished 11th.