The Sun says Blues chief executive Peter Kenyon jetted into Madrid to agree the transfer after the Brazilian right-back went on strike.
Alves pulled out of tonight's Champions League qualifier against AEK Athens to avoid being cup-tied for Chelsea's Euro campaign.
Alves, 24, claimed he was "psychologically" unable to play.
Seville wanted £30m but Kenyon haggled down director Ramon Monchi Rodriguez to £21.5m up front with another £3m linked to Alves' success. He will now travel to Stamford Bridge for a medical.
Meanwhile, the Guardian says the Brazilian has told friends that he has already agreed contract terms with Chelsea.