Kone-Doherty, 19, leaves Liverpool just as his contract was due to expire at the end of the season.
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Liverpool will receive around €750,000 for the sale, though they have negotiated a bumper sell-on clause, says the Mirror.
Kone-Doherty will be available for the new season kickoff at Molde.
He posted to social media: “I am truly so excited to open this new chapter in my career at Molde — time to work even harder!
“I’d like to thank everyone at Liverpool FC for the last three and a half years for helping me develop into the player I am today.”
Kone-Doherty has signed a deal with Molde to 2030.