Zirkee's move from Bologna was confirmed on Sunday.
He was seen in conversation with Mainoo after England's Euros semifinal win against Holland and he told club media yesterday: "Yeah, he just asked if I was coming. I said 'see you soon after the final'."
Zirkzee also said: "Having had discussions with the manager and the leaders of the club I know how exciting the future is going to be here and I cannot wait to play my part in achieving success for Manchester United.
"I am a player who has always dedicated everything to winning; I'm ready for this next challenge, to go to another level in my career and win more trophies. It is a privilege to join such an iconic club. I have to take a short break now after being with the national team, but I'll be coming back ready to make an impact straight away."