I currently live in Israel, and I am planning to fly to America on December 2, and fly back on December 6. I have two passports: one American, one Israeli. The American passport is fine, but the Israeli expires on December 23. Will I have a problem?
The reasoning behind minimun passport validities as an entry requirement for many countries is that they wish to ensure that you will leave at the end of your permitted stay. If your passport expires during the time that you are allowed to stay for, then it may be problematic to get you on to a method of transport and in to your destination, especially if your journey would involve transfering in a third country. Sometimes the minimum passport validity is extended beyond the normal stay time - one presumes that is to ensure they can be rid of you even if you overstay your welcome.
As a passport holder for both the countries you intend to visit, you are entitled to remain in either state for as long as you wish, so enforcing a minimum passport validity would not make any sense.
However, Israeli rules state that as an Israeli citizen, you must enter Israel with a valid Israeli passport or other travel document. So, if you were delayed in the US by some unfortunate circumstance, you may find yourself needing to procure an emergency travel document to return to Israel. It seems you may still have time to renew your passport before travelling however, although you would have to face a considerable delay to face any trouble when returning home.
Other than that, make sure to follow the advice given to in "Which passports to use travelling between two countries of citizenship?" and "I have two passports/nationalities. How do I use them when I travel?"
In your case, your Israeli passport will still be valid when you come back. So clearly there is no issue.
In any case, as an Israeli citizen, you can come back to Israel even on an expired passport. It needs to be valid when you leave the country but can be expired when you fly back.
I know because (1) I asked at customs and (2) I did it with no issue.
When you enter the United States, as a US citizen you must present your American passport. And when you return home, your document will still be valid. So no, you shouldn't encounter any problem.
You should apply for a new passport before leaving Israel, however, in case something complicates your trip and you are not able to return when you expect to.