The Blues agreed a fee with Real for Spain international Morata on Wednesday, with 'The Independent' believing that the wheels will now be set in motion with regard to Costa's exit.
Costa has made it clear that he wants to return to former club Atletico Madrid this summer after being told by manager Antonio Conte that he did not feature in his plans for the upcoming season.
The deal had been complicated by Chelsea's inability to bring in a suitable alternative, with Romelu Lukaku being snatched from under their noses by rivals Manchester United.
The same source reports that Costa will be now free to move once Morata completes his move to the capital, with the Blues asking for £44m for the former Atletico Madrid man.