The versatile Zinchenko will be joining the Gunners, who like City are on a pre-season tour in the United States, with the formalities of the transfer still to be completed.
Zinchenko, who joined City during 2016 from Russian side Ufa, is expected to leave City's training base to link up with Arsenal, where he will work again with manager Mikel Arteta, who was assistant to Pep Guardiola at the Etihad Stadium.
Arsenal have moved swiftly to strengthen the squad during the summer window, with Brazil forward Gabriel Jesus having already joined from Manchester City in a reported £45million deal.
Midfielder Fabio Vieira was signed from Porto, Brazilian youth international Marquinhos joined from Sao Paulo and American goalkeeper Matt Turner also completed a switch to the Emirates Stadium.