The midfielder, who joined City from Aston Villa a week ago, appeared to jar his leg follwoing a challenge with Luka Modric during the International Champions Cup match against Real Madrid in Melbourne, Australia.
City signed the England international to bolster their options in the centre of midfield but could now be without their £8 million signing at least for the remainder of pre-season.
Delph was beset by injury problems earlier in his Villa career, missing most of the 2010-11 season with a serious knee injury and the second half of the following campaign with an ankle problem.
He shrugged off those early set-backs to play a leading role at the Midlands club in recent years, and was handed the captaincy by Tim Sherwood last season as Ron Vlaar struggled with fitness problems of his own.
After initially turning his back on a move to City and pledging his future to Villa, he made a dramatic u-turn and signed a five-year deal at the Etihad Stadium.
Shortly after Jesus Navas replaced the Delph at the Melbourne Cricket Ground, City fell behind to a smart volley by Karim Benzema.
Cristiano Ronaldo, linked to a return to Manchester United in Friday's newspapers, made it 2-0 a few moments later when he latched on to Toni Kroos' measured ball over the top to clip past Joe Hart.