The German, 29, has agreed a three-year deal at the Etihad Stadium following the expiry of his contract at Arminia Bielefeld.
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Ortega, who made 220 appearances during two spells at Arminia Bielefeld, is set to join Pep Guardiola's squad for the preseason tour of the United States.
"This is a fantastic move for me," he said. "Manchester City are an amazing team, a squad with world-class quality in every area.
"To be given the chance to join this group of players and help continue the club's success is a dream for me. "I have enjoyed my time in Germany and I want to thank the Arminia Bielefeld fans for their support.
But this new challenge of coming to City and playing in the Premier League is too good for me to ignore. I can't wait to get started, meet my teammates and begin working with Pep and his backroom staff."
Steffen, who has made 21 first-team appearances since arriving at the Etihad in 2020, is set to move on loan in a bid to find more regular first-team football and bolster his chances of starting for the U.S. at the World Cup, which begins in Qatar in November.