Carroll, who played for United between 2001 and 2005, saw their struggles first hand last week when his current side Olympiacos beat Moyes' men 2-0 in the first leg of their Champions League last-16 clash.
The Northern Ireland number one was not involved in the game, but was an interested spectator from the sidelines.
"It surprised me a bit, the way Manchester United played against us because, for me, they are a team who want to win every game," he told Press Association Sport.
"They haven't changed too many players from the side that won the league, but David Moyes has had a tough start.
"He seems like a nice guy so I hope he does well, but the fans will have to be patient. I just don't know how patient United fans can be because they don't want to do an Arsenal and not win a trophy for that long, or be like Liverpool and go without winning the league for so many years."
While United's defeat in Athens was the subject of much discussion in England, with many seeing it as the worst display yet since Sir Alex Ferguson's retirement, the game has attracted the same amount of attention in Greece for very different reasons.
"That night was probably the best Olympiacos has ever had in the Champions League," he said.
"I was there and it was fantastic. The fans were going crazy. The way things are financially in the country is very hard right now and this takes things off their shoulders a bit. All people talk about is the football and how we beat Manchester United.
"It's going to be another hard night when we go back to Old Trafford but if the club could reach the quarter-finals it would be a blessing for everyone in Greece."
Carroll, 36, is not registered for European competition having slipped down the pecking order since a coaching change last year, but he is still planning on making the trip to his old stomping ground.
"It will be nice to get back to Old Trafford and see some old faces, people behind the scenes who I've not seen in a long while," he said.
"It's always good to see those people, but of course I want Olympiacos to win - that's my team now."