Costa hit 20 Premier League goals for Chelsea in his first season in his first season in English football, having joined from Atleti last summer for £32m.
The second half of his campaign was beset by injury and suspension, however, and on Monday night reports emerged that he had told his family and friends that he was desperate to exit Stamford Bridge and return to the Vicente Calderon.
But speaking after Chelsea's 1-0 victory against Sydney on Tuesday afternoon, Mourinho moved to set the record straight.
"He’s the guy responsible for everything good we have in the dressing room," Mourinho said. "He’s so happy, so funny, he enjoys every minute and is the happiest guy in the dressing room.
"If this is unhappiness I’m happy with that. He was not happy with the news."
Costa featured for 40 minutes of the first half in the victory over Sydney, with Mourinho keen to give the fans in Australia a glimpse of the striker in action.
"Diego was another one that respected what we found [the support of fans in Australia]," he added. "We felt it when we arrived, in the hotel and at training. We wanted to try to play our best players. This group of 10 seniors, we played them all.
"Diego tried, we didn’t want to take any risks. I told him to give the fans at least half an hour and he played until the 40th minute, we felt that was enough."
Costa racked up 20 league goals but failed to hit the target in any other competition, including the Blues' run to the League Cup final, where they beat Tottenham to lift the trophy, or in seven Champions League outings.