This should be okay in both directions.
Rules that require additional future validity of passports never apply for entering the country that issued the passport. It would be pointless for them to do -- countries have such rules to make sure they can send visitors back to their home country, but that only works if the passport is still good for the home country all the way up to the date written on it. If Italy wanted the passport to lose power completely three months earlier, they would just have issued it with an earlier date of expiry in the first place!
(This works for the entire Schengen area, by the way. The usual three-months requirement applies only to third-country passports in the first place).
As for going back to the US, there are three independent reasons why you should have no problems:
- Passports issued by Italy are exempted from the "six months or remaining validity" requirement in the first place.
- The six-month rule only applies to non-immigrant visitors, not to permanent legal residents. (8 U.S.C. §1182(a)(7)(B)(i)(I)).
- You can enter on your green card alone without showing the Italian passport at all.