The Ghana international joins City after they agreed to pay the £65m buyout clause in his Bournemouth contract.
Semenyo has signed a contract until 2031 and will wear the number 42 shirt previously worn by Yaya Toure.
"I am so proud to have joined Manchester City," Semenyo said.
"They have set the highest of standards and it's a club with world-class players, world-class facilities and one of the greatest managers ever in Pep (Guardiola).
"I have so much scope for improvement, so to be at this club, at this stage of my career, is perfect for me. It's a real privilege to be here.
"My best football is yet to come, I am sure of that."
He also stated: "I mean you want to be a part of the team, to be honest.
“You want to be winning those trophies, you want to be playing with the best players in the world, so that perception is that you just want to be involved.
“I thank God that now I’m involved and we can win many more trophies.”
Semenyo was playing for Bristol City three years ago and asked about his personal development, he credits his faith for this stunning rise.
“It starts in the mornings when I’m praying, day and night I make sure I always pray and make sure I always write down things where I can improve and get better,” he said.
“Having the right people around me, I feel like those three things have got me the success I’ve had to day.
“So, I just need to keep them around me, keep improving, keep writing things down and keep praying.”