Draxler, 20, has been attracting interest from a host of top European clubs after breaking through into the Schalke first-team in 2011 helping his side qualify for the Champions League last season.
However the attacking midfielder is growing increasingly frustrated by being played out of position by manager Jens Keller and voiced his concerns earlier this week.
According to reports in Germany, Chelsea are seeking to end Draxler's frustrations by offering out-of-favour winger De Bruyne as part of a deal to bring him to Stamford Bridge.
Chelsea hope to trump all other interested parties by allowing De Bruyne return to Germany, where he impressed during a spell with Werder Bremen in the Bundesliga last season.
The Belgian winger, who is yet to complete a full 90 minutes for Chelsea since his 2012 move from Genk, has been in talks with Atletico Madrid over a move to Spain.
However Jose Mourinho is hoping to wrestle De Bruyne's exit back into his favour to secure the services of Draxler in the upcoming transfer window.
Atletico boss Diego Simeone is desperate to ensure De Bruyne arrives at the Vicente Calderon Stadium with star man Koke expected to depart in the winter window with United leading the chase for his signature.