The 24-year-old has agreed to a five-year deal with the Merseyside club but McMahon says there's no guarantee that the former Southampton defender will be an instant success.
"Clyne is a good, young English player with a bright future, but he will find out quickly that Liverpool is a very different kind of club to Southampton," Malaysia-based McMahon told ESPN FC.
"It's like a goldfish bowl at Liverpool while at Southampton you can a couple of iffy games and nobody pays that much attention."
McMahon added that Clyne will need to get forward and help Liverpool score more goals, which was a problem for them last season after the departure of Uruguay international Luis Suarez and injury-plagued campaign of England forward Daniel Sturridge.
"Clyne will be expected to contribute a lot in an attacking sense while taking care of his defensive job," McMahon said. "He will need find his form quickly because Liverpool must start next season well."
Clyne is Liverpool's sixth signing of the summer and will join former Southampton teammates Adam Lallana, Dejan Lovren and Rickie Lambert at Anfield.
"I think it's strange that so many Southampton players have joined Liverpool in the last year or so," McMahon said. "Why can't Liverpool own scouting system find these kind of players?"
McMahon added that he was looking forward to seeing Livepool's new-look squad in the game against the Malaysia Select XI at the Bukit Jalil National Stadium on July 24.
The match is organised by Pro Events and tickets are available at www.ticketpro.com.my.