The 20-year-old was given his big chance to impress when introduced as a substitute against Arsenal in the Premier League last May, before being handed a start in the final-day win over Crystal Palace.
Speaking ahead of Wednesday's home EFL Cup clash against Burton Albion, which he is expected to start, the central midfielder told reporters: "It's crazy when you think back all those years to when you're five years old going to the development centre, and then going all the way through the ranks.
"It honestly is a dream come true for me to be playing in the first team, and playing against teams like Real Madrid [in pre-season]. But obviously you just want more and more of that and that is the main thing for my career - to be playing in big games, in competitive games and just do my best and hopefully see where it takes me."
Asked what it is like to work under Mourinho, the Englishman added: "He's top - he's perfect for a manager and that's why he is manager of Man Utd. He's a really good guy.
"Things he does say stay between me and the manager but a lot of it is help and advice, really. It's all like when we go out on the pitch, what I can do better, things that I can improve on and he is brilliant for that with young players."
McTominay, handed the number 39 shirt for this season, has been a part of the United setup since 2002.