The defender played no part in the victory at Villa Park, having suffered an ankle injury during Team GB's defeat at the hands of South Korea at the quarter-final stage of the Olympic men's football tournament.
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"With Micah, we thought it was a small injury but instead he will be longer," said Mancini.
"He has a problem with his ankle, a big problem maybe, and it may be one month out."
The injury means that the 24-year-old defender can take no part in England's friendly on Wednesday night against Italy.
City team-mate Joe Hart also had to pull out of the squad as well as missing the Chelsea game with a back problem, but the 25-year-old's condition is not worrying Mancini.
"Joe had a problem with his back, but I don't think it is serious," he added reassuringly.