Serie A outfit Inter are reportedly in talks with Tottenham to bring soon-to-be free agent Eriksen to San Siro in January.
Eriksen – who is out of contract at the end of the season – is keen for a new challenge, having joined Spurs from Ajax in 2013.
Asked about Eriksen after he was booed by Tottenham fans during Tuesday's 2-1 FA Cup third-round replay win over Middlesbrough, Mourinho told reporters: "He played very well. Played very well, very professional which is what I expect from him.
"If his decision is to leave, I think he has to leave with his head up, if he gives everything which is what he tries to do for the team.
"Fans, it's always to respect, we have to respect but I think the boy did it for us today."
While Eriksen's future appears away from Tottenham, Benfica midfielder Gedson Fernandes is reportedly close to arriving on loan from the Portuguese giants.
Amid talk of an initial 18-month loan deal, Mourinho said: "No news. I didn't want to know anything during the day.
"During the day it was just about the game. He doesn't have news for me so nothing."
In the FA Cup replay, Middlesbrough's hopes of an upset were severely compromised within two minutes of the replay when goalkeeper Tom Mejias' loose pass was intercepted by Giovani Lo Celso, who capitalised with a composed finish into the bottom-left corner to give Spurs an early lead.
Davinson Sanchez overhit his own pass from the back and then lost his footing, allowing Lukas Nmecha to turn inside Japhet Tanganga and draw a fine save from Spurs goalkeeper Paulo Gazzaniga.
Boro were quickly made to regret not capitalising on that opening when Jonny Howson lost possession and Erik Lamela eased beyond Paddy McNair and Dael Fry to double the Tottenham lead.
The away side were able to make the match a nervy finish when George Saville was allowed to advance and slot past Gazzaniga's apologetic dive, but Tottenham held on to progress to the fourth round.