Arsenal spent £45m on Jesus and have also bought midfielder Fabio Vieira, goalkeeper Matt Turner and young winger Marquinhos.
Zinchenko’s likely arrival is expected to accelerate the departure on loan of left-back Nuno Tavares, who deputised for Kieran Tierney at times last season.
Arsenal are also looking to find buyers for a number of fringe players including Bernd Leno, who is wanted by Fulham, and midfielder Lucas Torreira, who has been targeted by teams in Italy and Spain.
City are now expected to step up their pursuit of Brighton left-back Marc Cucurella, who was identified as their first-choice left-back target earlier this summer.
Brighton are thought to want around £50m for the Spanish left-back, who only moved to the Premier League in August last year. Brighton signed him from Getafe for a fee of around £15m, and will only sell if their valuation is met.