The teenage midfielder has signed an 18-month deal and will move immediately to the Amsterdam ArenA.
Ajax`s Scandinavian contacts rate the youngster highly, and have secured an option for a further two years should he impress in Holland.
Lindgren, 18, had a trial with Ajax at the end of last year, and will predominantly work with the juniors under Danny Blind and John van't Schip`s A2 side.
In other news, Koeman (pictured) has promoted Stefano Seedorf and Wesley Sneijder to the first-team.
The former Dutch international has been impressed with the pair`s progress.
Seedorf, nephew of Milan star Clarence, played for the first-team recently, while Sneijder is seen as one of the brightest midfield talents in Holland.
He has been tipped as a future Dutch international and his promotion is seen as a natural progression.
