Israel is to introduce a resolution to the United Nations General Assembly for the first time ever.
Annan has criticised both the General Assembly and Israel
The resolution - calling for Israeli children to be protected from violence by Palestinians - is Israel's way of testing the waters at the UN.
The world body, which has a large Muslim and Arab contingent, passes more than 20 resolutions criticising Israel each year.
UNGA resolutions are not binding, unlike Security Council measures.
Israel's relations with the United Nations are tense and often openly hostile.
The country ignores the dozens of resolutions the General Assembly passes condemning it each year.
But Israel decided to introduce its own resolution, which seems closely based on an Egyptian one in defence of Palestinian children, after a suicide bomber killed 21 people in Haifa last month.
"After the attack in Haifa, we can't just let it be," Ariel Milo, a spokesman for Israel's UN mission, told the Reuters news agency.
Four of the dead in the Haifa attack were children.
Israel says it would prefer that neither the Egyptian resolution nor its own win approval - or, failing that, that both are passed.
The US blocks anti-Israel resolutions in the Security Council
"The test will be if they pass the Palestinian one but not ours," deputy Ambassador Arye Mechel told the Associated Press.
The UN has not always been so critical of Israel - in fact, the country was created by a UN vote in 1947.
The UN moved to partition the British mandate of Palestine into two states, one Jewish and one Arab.
The Arab part was divided among Israel, Egypt and Jordan in the wake of war in the region in 1948.
A solid anti-Israel bloc emerged at the UN after Israel's victory in the 1967 war left it in control of swathes of Arab territory.
In 1975, the General Assembly passed a resolution equating Zionism with racism, severely rupturing relations.
The resolution was repealed in the 1990s but many Israelis still view the UN with bitterness.
The United States blocks anti-Israel resolutions at the Security Council if Washington does not consider them "balanced".
Un Secretary General Kofi Annan has criticised both the General Assembly's anti-Israel resolutions and the present Israeli Government.