The 25-year-old midfielder pulled up in the first minute of Tuesday's game at Wembley, which England won 2-0.
City manager Manuel Pellegrini said of Delph's prognosis: "I don't know how long it will be, we'll see.
"I think at this moment it will be a month and a half."
Delph signed a five-year contract at the Etihad Stadium in July - less than a week after saying he would stay at Aston Villa, where he was captain.
He also picked up a hamstring injury on his pre-season debut for City, during their game against Real Madrid in the International Champions Cup in Melbourne.
Delph returned to action as a substitute in his side's Premier League win over Everton on 23 August and also came off the bench in their victory over Watford before the international break.
Manchester City's Fabian Delph is expected to be out for six weeks after picking up a hamstring injury playing for England in their Euro 2016 qualifier against Switzerland.