Both clubs announced the transfer involving the former Liverpool star which German media estimated at around 30 million euros with an annual salary for Mane of 40 million plus 10 million in results-based bonuses.
Meanwhile , Chelsea have signed teenage French midfielder Lesley Ugochukwu from Rennes on a seven-year deal for 27m euros (£23.2m).
Ugochukwu, 19, has made 47 Ligue 1 appearances for Rennes and has also featured in the Europa League.
He is yet to make his senior debut for France but has played for Les Bleus at under-17, under-18 and under-19 levels.
His arrival comes after Chelsea boss Mauricio Pochettino recently said the Blues needed to add to their midfield.
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