The current England manager was sacked after less than a year with the Reds, though whether Moyes faces the same fate at Old Trafford is unclear.
Still, Carragher feels there are some worrying signs at Man United that remind him of Hodgson's plight.
"Watching United labour against Olympiacos brought back memories of the situation Roy Hodgson experienced in his brief stay at Liverpool in 2010; the shock home defeats, the unexpected cup exits and certain players letting the manager down badly," he wrote.
"If a manager loses his players, there is often no way back and an issue Moyes has to contend with at the moment is the fact he is the focus of the criticism; United’s players will know this and can hide behind that, rather than looking at their own shortcomings. Again, that happened with Hodgson."
And the similarities go even further according to the former Liverpool defender, with Robin van Persie's current predicament also bringing back memories of Fernando Torres for Carragher.
"The way Robin van Persie, for instance, is playing reminds me of the way Fernando Torres was with Hodgson. His goal record might be good, as was Torres’ for Hodgson, but the quotes that he has made and the stories surrounding him make it feel like Van Persie isn’t behind Moyes.
"He didn’t miss a game last season in the Barclays Premier League but this year he has been in and out – is he thinking about the World Cup? – while his demeanour has served only to increase the perception there are problems. Torres did exactly the same."
Does Moyes remind you of Hodgson during his time at Anfield?
image: © Jenny Leigh