Zirkzee left Bologna on Sunday for United.
"I have a lot of trust and a good feeling about this step," he told manutd.com. "It wasn't a difficult decision. I mean, I love Bologna a lot, but Manchester United just gave me a very, very good feeling.
"I'm confident that what's planned will also happen.
"It's been a rollercoaster, the past few weeks. Some very good experiences and I'm very happy to be here. Obviously, making a transfer to Manchester United is something very positive.
"I just can't wait to get started and continue what I basically did last season. I'm just hoping to improve every day and do the best I can."
Zirkzee also said: "I'm very happy to be here.
"I mean, it's probably the biggest club in the world, so I can't wait to get started and get to know everyone and start this journey. I'm very, very excited.
"I’m going on a short break. Try and relax a little bit and then get back to Manchester and get it going.
"Obviously, it (Old Trafford) is an insane stadium. I can't wait to play in the stadium. Just very, very excited, very motivated."