Israeli Prime Minister Menachem Begin and President Reagan met for the first time yesterday in an atmosphere of great mutual warmth, but their initial encounter left unclear whether the cordiality will be translated into the closer strategic ties that Israel wants to strengthen its security.

After a day of elaborate ceremony and exchanges of compliments, Begin noted that he is on his 12th visit here and characterized it as "the warmest atmosphere I have ever enjoyed." But, he added in a pointed reference to his goals, he hopes his departure will see the word "alliance," as applied to U.S.-Israeli relations, transformed into "a practical term."

Begin, in seeking a U.S. commitment to greater military cooperation, is using as a bargaining lever the administration's proposal to sell an $8.5 billion package of jet fighter enhancements and radar planes to Saudi Arabia.

Israel contends that providing such equipment to a hostile Arab country poses a serious danger to its security. But, while the Israelis hope their supporters in Congress will block the sale, they also are using the issue as an argument for closer U.S.-Israeli military ties that would help to compensate for putting the advanced weaponry in Saudi hands.

Administration officials say privately that they agree in principle with the Israeli idea, and the president appeared to underscore that point when he told Begin that "we know Israelis live in constant peril."

However, the administration so far has been reluctant to talk about specifics, and senior U.S. officials have gone out of their way in recent days to discourage speculation that the United States might be ready to agree to proposals that the Israelis have suggested.

These include joint military exercises, use of Israeli territory and bases for American maneuvers, increased sharing of intelligence and storage of U.S. equipment in Israel for use in a Mideast crisis.

In response to questions about each of these possible steps, senior U.S. officials have insisted that no commitments have been made and that considerable further discussion and study is required.

That continued to be the case yesterday after Reagan and Begin met for two hours at the White House. A senior administration official said afterward that the time had been devoted primarily to allowing the two leaders to become acquainted and to a broad-brush exchange of views about the major issues troubling the Middle East.

The official, who declined to be identified, said Begin and his aides spent considerable time outlining the already familiar Israeli objections to the sale of the airborne warning and control systems planes, known as AWACS. The official said Begin stated his case "quietly and without rancor," and he added that Reagan, citing the broad-scale U.S. interests in the Mideast, reiterated his belief that the sale will be approved by Congress next month.

"No minds were changed," the official said, "but the differences were expressed under friendly circumstances."

Later, after a working luncheon given by Secretary of State Alexander M. Haig Jr., Begin said: "The president enabled us to bring all the facts we had at our disposal to prove it is a danger to Israeli security . . . . Each side expressed its opinion in serious and profound discussion."

Asked if the AWACS planes are "a mortal danger to Israel," the prime minister replied: "We don't use such dramatic terms. Israel will live forever. We believe it is a danger, and I said so to the president."

Replying to questions about whether his visit will produce increased military cooperation, Begin said, "Yes sir, we hope so. We all hope so." But he declined to go into specifics other than to say that the matter will be pursued intensively by Israeli Defense Minister Ariel Sharon, Haig and Defense Secretary Caspar W. Weinberger.

According to sources on both sides, other topics marked out at the White House meeting for intensive discussion during Begin's visit included future U.S. economic and military assistance to Israel, pursuit of negotiations on self-government for the Palestinian inhabitants of Israeli-occupied territories, and the civil war in Lebanon.

Begin met late in the day with Weinberger at Blair House. Defense officials termed the meeting cordial; they said the two men did not discuss the proposed sale of equipment to the Saudis.

Begin and Weinberger are to meet again on Friday, when they are expected to discuss such matters as storing military equipment in Israel. Defense officials said the Israelis expressed interest yesterday in selling additional equipment to U.S. military forces. There was no discussion of sharing spy satellite information, they added.