The Blues have not won a trophy since they lifted the FA Cup back in 1995, but Carragher, who played 737 times for the Reds, believes that the 20-year-old can inspire Roberto Martinez's side to glory.
"Barkley can become the boy wonder who scores winning goals in front of The Kop to settle derbies and inspire Everton to ending their trophy drought," Carragher wrote in his Daily Mail column.
"He is a special talent and could be the one to take over Steven Gerrard's mantle on Merseyside.
"Winning silverware has to be a priority for Roberto Martinez during his time at Goodison Park and supporters will have confidence he can do it, given what he achieved with Wigan last May, but there is the potential for much more."
Everton are currently fifth in the Premier League.