Morata has endured a hard spell in England since leaving Real Madrid 18 months ago, with his two goals against Nottingham Forest in the FA Cup his first strikes since a brace against Crystal Palace at the start of November.
Chelsea are happy to allow Morata to leave, but they first want to source a replacement, which they have identified as Gonzalo Higuain.
Several players have urged Morata to show more fight on the pitch - something his performances have always lacked - though their pleas did not get much of a response.
Costa started his career in Atletico Madrid's youth team before being signed by Real Madrid.
An agreement has already been reached between the Premier League club and Higuain, but they are looking to release him from his contract at AC Milan, where he is now on loan from Juventus. Very few who have watched him this season will be surprised.
'People expect a lot from Chelsea's number nine, ' said the Italian.
Wolves would hope to win the day by offering a Portuguese contingent to help him settle in England - which can be very tough for mid-season arrivals - and a greater degree of gametime than he might expect in competition with World Cup victor Olivier Giroud at Stamford Bridge.