I will say to friends, family, or other closer persons "お腹すいた", "お腹がすきました" or "あ~、腹減った(はらへった)"
However, to "not close" person, inculding strangers and someone I respect, I would not say words meaning "I'm hungry" no matter how politely you say, because my being hungry is none of their business.
If I want to ask someone (not closer) for permission to eat,I personally would say
「実は朝から食事をとっておりません。さしつかえなければ、今の空き時間を使って、食事をしてきてもよろしいでしょうか?」
「そろそろ12時ですし、さしつかえなければ、食事にいたしましょうか」
The point is :
1). When you ask for permission:
- Your being hungry cannot be reason, because it is your personal feeling which can be selfish.
- The fact that you have not eaten anything since morning or that it's around noon can be a reasonable reason, because it is a fact and it is received that people should eat in appropriate time.
2). Other situations.
Maybe I won't say "I'm hungry" to persons (not closer), because it is none of their business and they won't be interested in whether I'm hungry or not.
※ Even to strangers, I wil say ”もしや、お腹がおすきではありませんか?"、"お腹がすいていらっしゃいませんか", because it shows that I am taking care of "their" business.