Coach Van Basten, along with assistants John van't Schip and Rob Witschge, head to the Amsterdam ArenA after leading Holland at Euro 2008.
The trio were all team-mates of Bergkamp when he was a player at the club from 1986 to 1993.
Van Basten has already offered Bergkamp the chance to study for his Netherlands Football Association at Ajax next season.
Van Persie, whose time at Arsenal overlapped with Bergkamp's by two years, feels the union could be a dream ticket for the Amsterdam giants.
The former Feyenoord forward told Sportsweek magazine "It is extremely positive, what's happening there.
"Van Basten, Van 't Schip, Witschge and Bergkamp are pure football men. When you train under them things can hardly go wrong.
"Van Basten and Bergkamp together would have so much to offer. I would like to play under such a combination - only not at Ajax of course.
"You could always ask Dennis things. Although he never came to you by himself, he helped everybody.
"Guys who showed a special drive were players who he put his energy into. No matter what was wrong with them, he looked through that."