The Ghana left-back, 21, has been earmarked as a replacement for Filipe Luis, who was sold back to Atletico Madrid after only a season at Stamford Bridge. Negotiations have been continuing for weeks, with the Premier League champions having a bid of £17.2m (€25m) rejected over the weekend.
Augsburg had been holding out for a fee closer to €30m but with Rahman having already agreed personal terms for his move to Stamford Bridge, the Bundesliga side have now reached a settlement over the structure of payments. Chelsea are expected to pay around £14m up front, with several significant bonus payments up to £6m due should he play more than 100 matches.
“I would like to continue working with all players, but when you sign someone like Baba from the second division and he’s on Chelsea’s radar 12 months later, it does make the coaching staff proud,” said the Augsburg coach, Markus Weinzierl.
Chelsea were alerted to Rahman’s promise by their former manager Avram Grant, who coaches Ghana. Rahman was a member of the squad who reached the final of the Africa Cup of Nations in February.