when Israeli jet fighters, including U.S. built F15s, shot down nine Soviet-built Syrian Mig21s in air battles over Lebanon in 1979, a key to the Israeli victories lay in a small, ungainly looking plane that was circling out of sight far from the battles.

The plane was also built in the United States, by Grumman, and is known as the E2C Hawkeye. It is a radar surveillance aircraft, much smaller than the four-engine E3A AWACS surveillance jets the United States has agreed to sell Saudi Arabia.

But it does the same kind of job, using electronic eyes to spot approaching enemy aircraft 200 miles away.

It is, in large measure, because Israel knows what Hawkeye can do that it is most worried about the administration's controversial proposal to sell five Boeing-built Airborne Warning and Control System jets to the Saudis beginning in 1985.

The Hawkeye radar, according to informed sources, picked up the Syrian Migs as they were rolling down the runway, and the Israelis were waiting for them. A new brochure from Grumman extolling the Hawkeye confirms this, reporting that "Syrian Migs are constantly under surveillance even while taking off from bases near Damascus." Sources say the Hawkeye radar can spot anything going more than 80 mph.

Pentagon officials briefing reporters last month played down the liklihood that AWACS could become a threat to Israel and stressed its importance to the Saudis in meeting challenges from other countries. Secretary of State Alexander M. Haig Jr. last week also urged congressional opponents of the sale to wait for final details before labeling it a threat to Israel.

But there are some U.S. military specialists experienced in both Middle East affairs and AWACS operations who take a different view.

They believe the plane could become a source of instability in the Middle East. It could even, they feel, once again catch Americans in an Arab-Israeli shoot-out, as happened in 1967, when Israeli ships and planes shot up and almost sank the USS Liberty, a Navy intelligence ship that was eavesdropping off the Gaza Strip.

There is an obvious political debate over the proposed AWACS sale to the Saudis. There is also a parallel debate over defense issues, which Congress will also have to consider when it decides whether to approve the arms sale. in view of the defense specialists who are critics of the proposal, Saudi AWACS, with their ability to look hundreds of miles across borders, will heighten Israeli fears that their pre-eminent military striking arm -- their air force -- will be living in "a glass house."

In every recent war, the Israelis have used their air arm swiftly and effectively to neutralize enemy air forces. U.S. military officers say Israeli's concern for the security of its air force borders on "paranoia" and that this will probably get worse next year as Israel gives up its air bases in the Sinai desert under the peace treaty with Egypt and pulls back into more confined borders.

Thus, some specialists believe that if an AWACS were operating over Saudi territory anywhwere near Israel's borders during a crisis in the Middle East in which the armed forces of many countries are on alert and primed to fight, it likely would be one of the first targets hit by Israeli planes. If Americans were helping the Saudis operate the complex planes, it is conceivable that another Liberty-type incident could develop, they say.

Senior officials in both the State and Defense departments, however, believe this scenario is far-fetched.

If Israel launched a pre-emptive attack against AWACS, it undoubtedly would be with the help of the four Hawkeye planes the United States sold it in 1978. Hawkeye was originally developed for, and is in service with, the U.S. Navy.

AWACS and Hawkeye can keep track of each other electronically, although the AWACS radar has a somewhat longer range. Hawkeye, however, also has a passive detection system that would allow the Israelis to spot the AWACS radar beam even when the Hawkeye radar is turned off.

Beyond this potential Israeli-Saudi game of electronic cat-and-mouse, Egypt also has said it wants to buy the Hawkeye. The Egyptians, mindful of Libya and othrer hostile Arab air forces, believe it will add stability and defense capability for the region when they, too, have these planes.

But skeptics among the U.S. specialists believe the prospect of three different air forces flying in the same area with sophisticated radar surveillance planes supplied by the United States might put more of a hair-trigger on pre-emptive strikes during a period of high tension.

American experts familar with AWACS and Hawkeye say they are superior, especially AWACS, to anything of their kind in the world, including warning planes in service or under development in the Soviet Union and England. No other allied country has been given AWACS, although 18 planes are on order for the NATO command in Europe.

Skeptics believe the Saudi AWACS will not only make Israel nervous, but Iran and Iraq as well. They argue that these planes, which can stay in the air 11 hours, or double that with aerial refueling, could keep an eye on the region but be based on Diego Garcia in the Indian Ocean or in Oman in the Persian Gulf and kept under U.S. control.

Some U.S. military officers also fear AWACS or other advanced weapons will fall into the wrong hands, especially in Saudi Arabia, where there are large numbers of foreigners in a country that critics believe may be less stable than Iran seemed before its government was toppled.

These critics claim that the United States might have lost AWACS in Iran, as it did F14 fighters and Phoenix and Hawk missiles, had the planes been delivered under a request by the late shah of Iran that had been approved by the Carter administration. Last week, a dozen U.S. Air Force F15 pilots protested sending the latest U.S. Sidewinder missiles to the Saudis in the same AWACS deal.

The administration, however, believes that Saudi deal is of great importance. The Saudis want the planes, and the United States wants to maintain good relations with its main foreign oil supplier. The sale also reduces the cost to the U.S. Air Force of the 35 AWACS planes it is buying.

If the entire arms deal proposed by the Reagan administration for the Saudis, including 62 F15 fighters with new missiles and aerial refueling tankers, is approved, the United States will have a stockpile of front-line weapons in the Persian Gulf. And they could be used jointly with the Saudis to ward off any major threat to the oil fields, including moves by the Soviets, supporters argue.

The administration says that with Iran now in unfriendly hands and with Soviet troops in Afghanistan and Moscow's military influence in the neighboring Horn of Africa region, it is vital to beef up Saudi defenses.

In a speech to the Foreign Policy Association in New York last month, Saudi oil minister Sheik Ahmed Zaki Yamani put it this way:

"If you look at the map and monitor Communist movements around the oil fields -- whether the Horn of Africa, South Arabia, in Afghanistan and their activities in Iran -- you will know that the aim is to surround the oil fields and you will know that they won't attack directly because this would be the third world war."

In effect, he was saying such an attack could come from any direction by Soviet surrogates.

But Yamani also said Israel is "a far more tangible" threat than Moscow and that creates a problem for Pentagon officials who have been trying to explain why AWACS is no threat to Israel.

These officials say it would be suicidal for the Saudis to send their AWACS near the northern borders of their vast country from where they could see all of Israel. The planes would be sitting ducks for an Israeli attack that would come so fast the Saudis would not be able to use the date the surveillance craft collected, the Pentagon says, or pass it on to other Arab air forces.

Such coordination, the U.S. maintains, takes years of practice and training and special equipment that will not be supplied.

The strategy surrounding by U.S.-Saudi arms deal is to provide forward defense, meaning the ability to shoot down attackers before they could reach the vital oil refinery and shipment area of Ras Tanura on Saudi Arabia's east coast.

AWACS is meant to provide both early warning and a command post to guide the F15s, with the new Sidewinder missiles that can attack an enemy head-on, so interception can take place as far away from the oil facilities as possible.