Yes it is possible. However it is also possible that the request not to have the passport stamped will not be honoured and your passport will be stamped anyway. If such a stamp will create a major hassle when you get back home, it is not worth taking the risk. However a few Gulf states (e.g. UAE, Oman, Bahrain) don't care too much about Israeli stamps these days, so in those cases it won't be a problem.
Apart from that - what is your nationality ? Many countries issue secondary passport for travel to Israel and if you fly in and out there will be no trace of a visit there in your primary passport when you return home.