Mata reacted furiously to being substituted in Chelsea's 3-0 win over Southampton on New Year's Day.
The Chelsea forward did not shake Mourinho's hand after being withdrawn in the 53rd minute and he flung his arms up in the direction of the touchline, where the Blues boss was standing.
The 25-year-old, who has started just nine Barclays Premier League games this season, had to be calmed down by Michael Essien, who was sat on the bench at the time.
Mourinho claims Mata reacted that way because he was unhappy at the scoreline - which stood at 0-0 when the former Valencia man was substituted.
Mourinho does not want to lose Mata in the transfer window, and he says he would be happy to talk to the player if he is unhappy.
"I want to keep him. I don't want him to go. That is my opinion, my wish, but my door is open," the Chelsea manager said.
"The club's door is open too so when a player wants to speak to us we are there waiting for them.
"If you are asking do you want the club to sell him? I don't want to."
Essien's future at Chelsea was thrown into doubt when his agent suggested that he could leave the London club due to a lack of playing time.
Mourinho, again, stressed he wanted to keep the player, but admitted that he could not offer the play guaranteed first-team football.
"I offer nobody - not Michael, not nobody (guaranteed playing time)," the Portuguese said.
"I understand that players think about national team, the World Cup and playing regularly, but my players have to think of the team before the players. Michael knows that perfectly."