Dolberg is currently in his first season with Ajax and has hit the ground running, with four goals from his first six Eredivisie appearances.
The 19-year-old Denmark Under-21 striker agreed a contract until 2021 earlier this year and is highly-regarded at the Amsterdam outfit.
Bergkamp spent seven seasons at the Amsterdam ArenA and two with Inter before joining the Gunners in 1995 and is now working as a coach at Ajax.
Dolberg has been compared to the former Netherlands international, and admits to being a student of the way his hero played the game.
"I have seen a lot of videos of my hero Dennis Bergkamp on YouTube," he said. "Dennis Bergkamp was a fantastic player and it’s great to work with him in training because he knows so much and he gives so many good tips and it stimulates you to listen more to the advice he gives. It’s a pleasure to be able to work with him."