The Belgium international enjoyed a successful loan spell at the Bundesliga side last season, but has started only two Premier League matches on his return to Stamford Bridge this term.
He was left out of the squad for Tuesday night's Champions League trip to Steaua Bucharest, with questions over his omission prompting an angry Mourinho to end his pre-match press conference early.
Bremen's general manager has revealed the club are in touch with De Bruyne's camp, and would willingly offer him an escape route in January.
Thomas Eichin told Germany's Bild newspaper: "Should Kevin be unhappy, we are an eager first contact.
"I am in contact with his agent and am informed about Kevin's situation at Chelsea."
The 22-year-old scored 10 goals in 32 Bundesliga games for Bremen last season.
"I know Kevin, if he doesn't play it is certainly a big problem for him," Eichin added.
"Let's wait and see what happens in the winter. We will be ready, if there is a possibility, to get him."