Earlier this week, Sky Sports said Arsenal would pay Premier League champions City 30 million pounds up front, with two million pounds in add-ons, for the 25-year-old, who will become their fifth recruit in the off-season.
Zinchenko worked with Arteta when the Spaniard was an assistant to Pep Guardiola at City from 2016 to 2019.
"There are still a few formalities to get everything completed but we're really happy to have him," Arteta told reporters after Arsenal's 3-1 win over Orlando City. "I know the player really well. He's an exceptional footballer.
"He is someone that is going to bring as well another competitive edge to that dressing room. He was a natural No. 10 early in his career and we converted him into a left back which can do a lot of things that we want in our way of playing.
"That versatility is something that is going to be important for the team because we have players in that position (left back) who are more specific full backs, so I'm really happy."
Arsenal beat Major League Soccer's Orlando to claim a third pre-season win, with Eddie Nketiah (23), Gabriel Martinelli (21) and Reiss Nelson (22) getting on the scoresheet for the English club, while Facundo Torres (22) scored the home side's only goal.
Arsenal will also take on Premier League rivals Chelsea in the United States before rounding off their pre-season preparations against Sevilla in London on July 30.
They begin their league campaign on August 5 with a London derby at Crystal Palace.
