The Italy international wears contact lenses when playing but has recently experienced problems with them because of some instances of conjunctivitis.
City hope the 22-year-old could be playing again within two weeks and have waited for the international break to book him in for the procedure. He will, however, miss the Azzurri's World Cup qualifiers against Bulgaria and Malta.
City first-team coach David Platt said: "I think he is okay, it is just a normal procedure. It is just a minor operation.
"The contact lenses are the issue when he gets it, conjunctivitis. He can't get his contact lenses in, so he can't really play."