The Stamford Bridge club are keen to add a striker to the squad this summer, and they've lodged an inquiry with the Turkish champions as to the availability of Yilmaz.
The 27-year-old forward impressed in the Champions League this season, scoring eight goals as Galatasaray progressed to the quarter-finals. Yilmaz also top scored in the Turkish Super Lig, and hit 33 in 34 league games the previous season for Trabzonspor.
The report claims Galatasaray have told Chelsea that only bids of £20m and above will be considered.
Former Chelsea striker Didier Drogba has backed his Galatasaray teammate to shine at the highest level, even though Yilmaz has never played outside Turkey.
The Blues have started to look at other options in the transfer market as it becomes clear their top targets may not be achievable.
Radamel Falcao and Edinson Cavani are two to have been strongly linked with Chelsea, but the fierce competition for their signature this summer means Chelsea need to have a back-up in mind in case they fail to land a top tier target.
Cavani is wanted by Manchester City, Falcao by Monaco, while Robert Lewandowski is likely to join Borussia Dortmund teammate Mario Gotze by signing for Bayern Munich.
Because the club, who are expected to confirm the return of Jose Mourinho by June, are desperate to bring in another striker alongside Demba Ba and Fernando Torres.
Romelu Lukaku may yet return to Stamford Bridge to play a role in the first-team, but the Belgian international could still spend another season out on loan. Swansea are reportedly keen.