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The Chelsea forward did not shake Mourinho's hand after being withdrawn in the 53rd minute and he flung his arms up in the direction of the touchline, where the Blues boss was standing.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, but he admits 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."
Mourinho's decision to replace Mata and Andre Schurrle with Willian and Oscar turned out to be a wise one.
Oscar's deflected shot rebounded off a post into the path of Fernando Torres, who put Chelsea ahead.
Willian doubled the lead 11 minutes later and Oscar wrapped up the win eight minutes from time.
Southampton had held their own for the majority of the first half, but Oscar's trickery proved too much for the south-coast side, who have now won once in their past nine matches.
Oscar did blot his copy book when he dived in an attempt to win a penalty following the slightest of touches from Southampton goalkeeper Kelvin Davis.
Mourinho admitted Oscar deserved his caution, but insisted the Brazilian was not a serial diver.
"Oscar is a clean player," Mourinho said.
"He was waiting for the goalkeeper to come and smash him because that's what normally happens in those situations.
"It happened with Oscar at West Ham but the goalkeeper stopped before the contact so in the end instead of the goalkeeper trying to smash him it was (Oscar) trying to touch the goalkeeper. It was a fair yellow card."
The result keeps Chelsea within two points of Barclays Premier League leaders Arsenal, but Mourinho suggested his team are not contenders for the title.
"We are not candidates," he said.
"We are just candidates to win always the next match - and the next match is Hull City for the Premier League.
"I keep saying I know what I am building at the club. I know that next summer will be a moment for us to go to a couple of clinical buys to improve definitely our team in that direction but at this moment we work, we improve, we try to bring in people like Eden Hazard, Willian and Oscar to a different level of playing football and I think we are doing that quite successfully."
Southampton could have gone ahead had it not been for some last-ditch tackling from Ashley Cole, John Terry and Gary Cahill.
But their finishing let them down.
Talk of Saints pushing for a surprise European spot has diminished following their recent poor form, but Mauricio Pochettino is not a worried man.
"Rome wasn't built in a day," the Southampton manager said.
"We have been facing some of the top sides and that has been tough.
"We are a young side and we are still learning. We have to keep on believing on our style of play."