The striker will initially be registered with Warren Joyce's B team for the new season.
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Sillah, 19, hails from Gambia and joined Olympiakos only a year ago.
Forest's Head of Football Development & Talent Management, Craig Mulholland, said: “It’s really important our Emerging Talent pool has a blend of the best players from the local area who have come through our Academy, as well as top talent from around the world.
“Lamin is someone who has high potential, and we’re delighted to have added a goalscoring attacker to Warren’s young, exciting squad.
“He arrives here hungry to improve. We wish Lamin well for the year ahead and we are all looking forward to seeing how he settles into the Club.”