The striker is expected to move to Stamford Bridge for a fee of £32m, which constitutes the buy-out clause contained in his contract with the Spanish club.
It is understood that Chelsea first intended to complete the deal before the start of the World Cup, in which Costa played for adopted country Spain. However, a conclusion proved elusive last month and it is only now that Atletico have accepted the offer.
Costa will join Cesc Fabregas in moving from Spain to Stamford Bridge this summer, while veteran goalkeeper Mark Schwarzer has also signed a new one-year contract with the club. There was, though, confirmation last night that Ashley Cole has been released, alongside Frank Lampard.