The Brazil right-back signed a five-year contract at the Etihad Stadium, with City completing a deal believed to be worth £26.5million on Sunday.
After joining Madrid from Porto in 2015, Danilo won the Champions League in his first season with Zinedine Zidane's men, before helping them retain the trophy as well as win La Liga, the FIFA Club World Cup and the UEFA Super Cup last campaign.
City were beaten by Madrid over two legs in the Champions League semi-final in 2016 but now Danilo wants to deliver City's first European silverware since 1970.
Speaking to City's official website, Danilo said: "I'm really looking forward to the challenge and would love to win the Champions League with Manchester City."
Asked what influence vaunted City coach and former Barcelona boss Pep Guardiola had on his decision to join the club, Danilo replied: "He's been very important because I love his playing style and he's convinced me in the talks we've had.
"I'm looking forward to working with him. He's been one reason, but it's also that ambitious project of the club that convinced me to come."
Danilo is the second full-back City have signed in the transfer window following Kyle Walker's arrival from Tottenham. Left-back Benjamin Mendy is also set to join from Monaco after City allowed Aleksandar Kolarov, Pablo Zabaleta, Gael Clichy and Bacary Sagna to leave the club.
Given the expected competition for places between the new arrivals at full-back, Danilo's versatility is a bonus for City. And he confirmed Guardiola believes he can do a job at a number of positions.
"I don't have a favourite position," Danilo added. "I am used to playing as a right-back, but any time I'm on the pitch in the starting XI, I'm always happy.
"He [Guardiola] told me I'm ready and I can play in several positions, right-back, left back, midfield. I just hope to help him out as he expects."
On playing in the Premier League, Danilo said: "It is a competition I always liked, I always wanted. So I'm very excited to start on August 12."