The 22-year-old burst onto the scene under David Moyes during the 2013-14 season, but he quickly fell out of favour at Old Trafford and failed to make a single competitive appearance under Mourinho, spending last term on loan at Sunderland as they suffered relegation from the Premier League.
Januzaj completed a £9.8m switch to Sociedad last month as his six-year spell at United came to an end, and he is keen to rebuild his career at San Sebastian.
"Obviously, [Mourinho] is a great trainer but I did not have much of a relationship with him. I respect him a lot because he is the one who left the door open for me to leave Manchester United," he told MEN Sport.
"What I want now is to prove that I am a good player and that they were wrong to let me go. I want to grow with La Real and be a better player. That's my goal for the future."
United have included a buy-back clause in the deal to take Januzaj to Sociedad.