The former Borussia Dortmund star passed a medical and visited Carrington during his brief trip to England on Friday and is excited about the prospect of pulling on the red shirt, even though he is unsure which number he will be wearing.
"First, I went to the training facility [Carrington], then signed at Old Trafford," he told reporters at Narita International Aiport.
"Though it was a short stay, it meant a lot. It was a memorable day for me. There were many contracts to sign, which made me realise how big the club is.
"What came to my mind was how exciting and tough a challenge this is, and how scoring goals at this stadium would make me so happy.
"United is a fantastic organisation. For instance, the training facility is very well maintained. It is an environment I had never experienced before.
"I was excited to see the stadium. Having imagined myself playing there as my home stadium, I was impressed.
"Everyone welcomed me warmly and said 'Thank you for signing with us.' I know this is just the beginning but I am glad I have made it to the starting point."
Kagawa was asked if his shirt number was discussed following some talk he would wear the number seven vacated by Michael Owen, but the Japan star appeared to quash those rumours.
"We did [discuss it] but nothing will be confirmed until Sir Alex comes back," Kagawa