The Portuguese insisted during a press conference that he would be "delighted" if the Brazilian ends up staying, but refused to discuss speculation surrounding Luka Modric
The attacking midfielder's agent recently revealed that he was in talks with AC Milan over a possible return to San Siro.
However, the fee and the player's wage demands represent significant stumbling blocks, and Mourinho has explained that it is too early to tell if the proposed deal will go through, and only stated that Kaka will feature in Friday night's friendly against Benfica.
"He's going to play for 45 to 60 minutes, but I understand that you want to know if he is staying," the Portuguese told reporters at a press conference on Friday. "I do not know and neither do the club nor the player.
"But there is no doubt between us [Mourinho and Kaka]: he knows what I think and I know what he wants.
"The market is open until Aug. 31 and a lot can happen until then. But as long as he's a Madrid player, he will train with the first team and play with us.
"He will travel to the United States [for the club's pre-season tour] and if he stays, I'll be delighted, because we can still do things together."
While Mourinho gave very little away regarding Kaka's future, he was at least prepared to discuss the topic. However, he refused to be drawn at all on Madrid's pursuit of Luka Modric, declaring: "I do not talk about players from other clubs."
The former Chelsea boss was more forthcoming when it came to discussing reports that Nuri Sahin is on the verge of a loan move to England, but admitted all the same that he has already grown weary of the constant speculation surrounding Madrid's transfer dealings.
"I do not like this market that will last until Aug. 31," the two-time Champions League winner confessed. "It allows room for many questions. Not among us [the staff and the players], but among journalists.
"I want 20 players, not including goalkeepers, and to leave a door open for the youngsters. So, someone has to leave.
"I try to be honest and they know who they are. Nuri is living in a grey area, neither black nor white. But if there is an interesting possibility to play elsewhere, he would be better off leaving because we saw last year that he did not get many minutes here."
Mourinho arrived in the Spanish capital during the summer of 2010, fresh from his treble-winning campaign with Serie A giant Inter.