Costa, 26, has been one of the best goalkeepers outside Europe's top five leagues for some time, spending his entire career so far at Porto and establishing himself as first choice for Portugal's national team along the way.
He carries a €60m release clause, and while Juventus and PSG have both been credited with interest in the past, neither club has moved to make that interest concrete in recent weeks, instead turning their attention elsewhere in the market. Costa has made 246 appearances for Porto, keeping 108 clean sheets, and has been capped 48 times by Portugal, playing a leading role in the club's 31st league title last season with 26 clean sheets in all competitions.
Chelsea are the club now driving talks forward, with journalist Pedro Almeida reporting that negotiations between the two clubs have advanced significantly, though no agreement has yet been reached and it remains to be seen whether an official bid follows; other reports suggest Chelsea's opening offer could be worth around €40m, some way below the full release clause.
Interest comes as Chelsea continue to reshape their goalkeeping department: Robert Sánchez has struggled to convince as first choice, and Filip Jörgensen is separately closing in on a loan move to Strasbourg, leaving a clearer path for a new arrival between the posts.
Costa would represent a significant statement of intent from Xabi Alonso, whose build-up-focused approach would suit a goalkeeper comfortable with the ball at his feet, and would also see off rival admirers including Real Madrid, Atlético Madrid, Inter Milan and AC Milan, all of whom have been linked with him in recent months.
The move would come on top of a summer in which Chelsea have already brought in Danny Welbeck, 35, and Morgan Rogers, 24, as they continue to strengthen their squad under the new manager.