UNITED NATIONS — The Middle East peace process is stalemated in part because of disagreement over the appropriate land borders for Israel and a future state of Palestine. Now the Palestinians are taking the first steps toward establishing what their state would claim at sea.

The Palestinian ambassador to the United Nations, Riyad H. Mansour, said Friday that his government had begun preliminary negotiations with Egypt to define the extent of Palestinian-claimed territory in the Mediterranean off the roughly 25-mile-long coast of Gaza, the strip bordering Israel and Egypt where roughly 1.8 million Palestinians live.

Mr. Mansour said the negotiations, which he described as having begun recently, were possible because of the Palestinian territories’ United Nations status as a nonmember observer state, which the General Assembly recognized in November 2012.

That status empowered the Palestinians to achieve membership in United Nations bodies and treaties, like the convention that created the International Criminal Court, much to the irritation of Israel and the United States, its most important ally.