Barkley scored his first Chelsea goal in their 3-0 win over Southampton on Sunday afternoon, and his form in the early stages of the campaign has led to a return to the England squad.
Barkley will be looking to make an impression in England's upcoming UEFA Nations League clashes with Croatia and Spain, and Sarri has said that he is "really happy" with the 24-year-old.
"I am really happy with him. He has improved week by week, day by day and match by match," Sarri told reporters. "He had a very serious injury in the last two seasons, he is improving day by day and I think the best will come.
"I said from the first day that he has great physical and technical quality. But you have to improve from the tactical point of view and now he has improved I think.
"Potentially, he is a great player. He has quality - very important quality from the physical point of view. From the technical point of view, he is improving tactically. He will become a very important player, and for the England national team."
Barkley joined Chelsea from Everton during the January transfer window.