The Finn joins Mallorca as a free agent and has signed a deal to 2027.
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While with Chelsea, Bergstrom gained experience on loan at Peterborough and Brommapojkarna (Sweden), as well as training with the Blues' first team for the last two seasons.
Upon signing terms, he said: "I'm very excited. I just left Chelsea and I'm happy to be here. I can't wait to meet everyone and get started. I hope to help the club grow.
"I arrived there (Chelsea) at 16 and spent almost seven years at a fantastic club. I spent the last two seasons working with the first team: it was a crucial period in my development."
Bergstrom added, "I hope for a warm welcome. I'll give everything for this shirt. I'm here to give my all every day."