Goal revealed last week that the 23-year-old had a deal in place to link up with the reigning Premier League champions.
The initial terms of the agreement include a $7.5m (£6m) fee that could rise to $10m if Steffen reaches certain incentives.
Steffen told the Crew’s official website on seeing a high-profile transfer put in place: “Any success that I have had as a goalkeeper in MLS with Crew SC is a testament to those around me who have pushed me – team-mates, coaches, staff, family, friends and supporters.
“Today’s announcement regarding next summer is a special moment for me and I look forward to finishing my time with this club on a strong note. I am eager to continue to put in the work to improve and be the best teammate I can.”
Steffen will officially become a City player when the 2019 Secondary Transfer Window opens on July 9, 2019.
“This is a historic transaction for the club and a special moment for Zack Steffen that we are immensely proud of,” said Columbus Crew’s interim general manager Pat Onstad.
“Today’s announcement is the culmination of two-and-a-half years of development and success for Zack.
“As someone who worked closely with Zack throughout, on behalf of the club, I would like to say that we couldn’t be happier for him.
“Manchester City Football Club is one of the top clubs in the world and we are pleased to finalize this agreement that benefits the player and both clubs.
“We continue to believe that players that come to Crew SC have an opportunity to develop and become great players either here in MLS or abroad.
“Ahead of Zack’s departure next summer, our scouts already have players targeted as the club identifies a high-caliber replacement.”
Steffen was voted Major League Soccer Allstate Goalkeeper of the Year for 2018, becoming the second-youngest recipient of that prize and the Crew’s first since 1997.
He made his senior debut in 2017 and is considered to be a hot prospect for the future.
It does, however, remain to be seen what City’s plans are for him, with it likely that a loan move will be required as he waits on a UK work permit.