Fotbollskanalen reports the Finland U21 goalkeeper was in talks with IF Elfsborg, but instead is on his way to IF Brommapojkarna.
BP are now close to picking up the 21-year-old from Chelsea in a free transfer.
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Bergstrm is on a Chelsea contract that expires in the summer.
Chelsea signed Bergstrm from Finnish side TPS in January 2019. He is yet to make his senior debut for the Blues.