Steffen was signed by the Premier League champions in December but remained on loan with Columbus Crew for the first half of the Major League Soccer season last term.
He made his 84th and final appearance for Columbus in last month's 2-2 draw with New York City FC, notching up 26 total clean sheets during that time.
"I'm really looking forward to returning to Germany and becoming part of such a traditional club," he told his new side's official website.
The 2018 US Male Soccer Player of the Year and Goalkeeper of the Year was first-choice stopper for the United States at the CONCACAF Gold Cup, which ultimately ended in disappointment with a 1-0 defeat to Mexico in Sunday's final. However, on a personal level, the ‘keeper was applauded for recording four clean sheets during the tournament.
"Zack is a talented goalkeeper with great potential and someone who has performed on the biggest stages for the USA youth teams,” USA coach Gregg Berhalter told the Bundesliga official website.
Confirmation of Steffen's departure came on the same day that City loaned fellow goalkeeper Aro Muric to Championship side Nottingham Forest.
Muric will also spend the whole of next season away from the Etihad Stadium, but he signed a renewal before departing that keeps him under contract until 2024.
"Extending my current deal with City by a further three years is incredible and an honour for me," he told City's official website.
"I really enjoyed the 2018-19 season and to be involved in our successful EFL Cup run was an amazing experience.
"The players and supporters have made me feel so happy and part of everything and the club have showed their belief in me with this deal.