The 36-year-old has led the side to two promotions during two spells with the Cherries, guiding them to the second tier for the first time in 23 years.
Fletcher, who retired last year, told BBC Radio Solent: "I do believe he could be a future England manager.
"I speak to people all the time through the football world who tell me about their mangers, and for me Eddie is head and shoulders above anyone.
"I believe in him, I don't think there are many better managers around. He is so good at what he does. Yes, he has only been in football as a manager for four or five years but it's fantastic to see how much he has come on."
Bournemouth currently sit 12th in the Championship.