The 20-year-old has been linked to a number of clubs throughout the season due to an impressive campaign under manager Roberto Martinez.
The midfielder has also been tipped to travel to Brazil with the England squad for this summer's World Cup.
"I would like to see him stay at Everton for the next two or three years, and maybe try to win a trophy there," Carragher told talkSPORT.
"Eventually though, he'll have to play at the highest level and that's the Champions League. Everton are just knocking on the door but I'm not sure whether they can sustain that."
Everton are currently four points behind a Champions League spot in the table with a game in hand on fourth-placed Arsenal, whom they face this Sunday.