Israeli Defense Minister Ezer Weizman is coming to the Pentagon Sept. 14 with a demand that presents a ticklish problem for President Carter and an enticing plum for American airplane companies.

Israel, Wiezman is prepared to stress to Defense Secretary Harold Brown, wants to build its next fighter plane on its home ground -- American designed or not.

The leading candidates for Israeli home production are the F16, a single engine jet made by General Dynamics, and the F18, a twin-engined fighter manufactured by the team of McDonnell Douglas and Northrop.

Although Israel, as a third alternative, could choose to build a new fighter on its own, Pentagon officials insist this is a remote possibility.

As good as Israel has proved to be at modifying American weapons for Mideast warfare, they said, building a complete fighter plane would be too difficult and expensive for the Israelis to attempt at this time.

Pentagon officials expect the argument to be over which American plane and how much of it will be build by Israel's gaint aerospace firm, Israel Aircraft Industries Ltd. The company employs about 22,000 people and will be running low on work soon, according to defense officials.

Israel Aircraft Industries Ltd. is bueiling the Kfir fighter-bomber. However, its airframe came from a French plane and its engine, the General Electric J79, from the United States.

Israel has been frustrated in its attempts to sell the Kfir abroard, including United States refusal to allow sales to Bcuador, and wants to build a more sophisticated model to combat the Arab threat envisoned for the late 1980s and 1990s.

Weizman, Israeli sources said, will insist that building just part of the next fighter and assembling it is Israel will not be enough. Israel wants to participate in the whole process -- from blueprints to metal cutting to assembly.

While defense officials acknowledge such full participation would help Israel Aircraft Industries Ltd. to turn out sophisticated fighters, they said an undesirable side-effect would be what is called "the proliferation problem."

"Supposing," said one official in elaborating, "that 500 new fighters would have to be built to make it economically feasible to set up a production line in Israel. And supposing Israel needs only 50 or 100 of the planes. What happens to the rest of them?"

President Carter has pledged to curb the proliferation of sophisticated U.S. weapons around the world. Some U.S. officials believe that helping another country produce the latest weapons, with the risk of their being sold eventually to third countries, runs counter to that pledge.

Although the United States has entered into coproduction of airplanes contend helping Israel produce a hot fighter in the volatile Mideast would be a riskier undertaking.

However, turning Israel down on its request for joint production of a modern fighter might force Israel to go it alone, as difficult and expensive as that would be. The United States, if it had not supplied the engine or anything else for the resulting fighter, would not have the right to control sales outside Israel.

Wiezman is prepared to make that argument, sources said, when he meets Defense Secretary Brown. The betting within the Pentagon is that this will help focus the discussions on which U.S. fighter, and how much of it, the Carter administration is willing to let Israel manufacture.

Israel has previously sought to get production rights for the F16. Penatagon officials said Israel's request was not rejected, although the administration was not enthusiastic. The Israelis opted instead for getting the F16 fighters destined for the shah of Iran before he was deposed. That choice, according to the Pentagon, enabled Israel to obtain the first batch of the promised 75 F16s earlier.

Much to the distress of General Dynamics, which manufactures the F16, Israeli officials are exhibiting interest in buying the competing F18 fighter. Why they are doing this after ordering the F16 is the subject of much speculation inside and outside the Pentagon.

One U.S. government theory is that Israel is trying to play off one American contractor against the other to get the best offer. The next step would be convince the Carter administration to allow it.

Another theory is that the Israeli Air Force considers the F18 more modern than the F16 and more reliable, partly because it has two engines instead of one.

Whatever the reasons, Israel's new demands for production rights are expected to dominate the Weizman Brown meeting next month. Defense officials predict it will be mid-1980 before the administration makes its decision.