Current manager Roberto Martinez is under increasing pressure at Goodison Park.
The Blues boss is under increasing pressure at Goodison Park after a second successive disappointing campaign from his side.
They currently sit 11th in the table and are unlikely to finish in the top half for the second year running, a position far below where a side of their talent should be.
Talk of who could replace the Spaniard has already begun, with the names ranging from Eddie Howe at Bournemouth to Jose Mourinho.
Among them is currently Manchester City boss Pellegrini, who will be a free agent in the summer, but Wright says that he is not the man they would need.
“You just hope now that they do make the right appointment,” he told BBC Radio 5 Live.
“Manuel Pellegrini is of the older statesman kind of manager and maybe they need somebody younger now, and that’s no disrespect to him.
“With the forward moves they are now making, I think they need a young manager that’s going to take them forward.”
Who could replace Martinez at Goodison Park should he be sacked is certainly a hot topic at present.
Despite what Wright suggests, there is no denying that Pellegrini would be a good appointment for the club given his record of managing big clubs in the past.
That sort of experience could be what the Blues need as they seek to become challengers at the right end of the table using Farhad Moshiri’s money.