The Blues have reportedly had a €21.4 million approach for the Brazilian rejected, which led the forward to declare that it would be "a dream" for him to play for the club.
While Mourinho was unwilling to confirm whether or not the Premier League leaders had put in such a bid, he welcomed Costa's enthusiasm.
"I am always happy when a player tells us that he would like to play for my club," the Portuguese told reporters.
"I am always happy so the reaction for Douglas Costa is the reaction of being happy with his comments and every Douglas Costa in the world that would like to play for Chelsea ... these kind of feelings are always welcome."
Two players whose futures could lie away from Stamford Bridge are Mohamed Salah and Andre Schurrle and Thegoalmac understands that Salah has agreed a loan-to-buy deal with Serie A giants Roma.
While insisting that he did not expect either to depart before the Blues' FA Cup clash with Bradford City on Saturday, Mourinho indicated that a good offer for either player would be considered.
"The process has to be different," he remarked. "I think first of all what has to arrive at the club is a great proposal for the player to have the right to say I want to go, that is a consequence of this fantastic offer on the table.
"And then [Chelsea] comes to me and tells me of this offer on the table, which is very good for the club in financial terms, and after that we try to do with it, not the other way around.
"I don't know [if anyone will leave]. In this moment, no. I go day by day, match by match and, in this moment, every player is available. We don't have offers on the table to make us think that this player is leaving tomorrow before the game."
