Secretary of State Alexander M. Haig Jr. initiated a new U.S. push for early and visible progress in the Middle East peace process in a round of meetings with Israeli leaders here today.

According to U.S. and Israeli officials, Haig presented U.S. "ideas" for bridging the gaps in several areas between Israel and Egypt in the negotiations about Palestinian autonomy in the West Bank and Gaza.

The diplomats described Haig's offerings as suggestions of a general nature rather than formal or detailed proposals. After a four-hour meeting tonight with Prime Minister Menachem Begin, Haig himself termed them "general formulas in certain narrow areas."

Israeli Army radio said the suggestions dealt with land and settlement policy as well as the legislative powers of the proposed self-governing authority for the Palestinians. Official briefers of the two sides would confirm only that the scope and authority of the autonomous council was involved.

There was no expectation that Israeli officials would immediately agree to Haig's suggestions or that Egyptian officials would do so when the secretary of state travels to Cairo Thursday.

The hope expressed by U.S. officials was rather that Haig's ideas would bring about a common concentration on areas where major progress can be made in weeks to come, dispelling concern that the Camp David peace process is dead except for the physical return of the Sinai scheduled for April 25.

"Haig told us today he very strongly believes it is possible to reach an agreement," said an Israeli official. He added that Haig did not make clear how extensive such an agreement might be or when it might be reached.

The discussions were described as notably different in tone and form from Haig's talks here two weeks ago. On the earlier visit Haig was asking questions, according to the Israelis, but "now it is an exchange of views on how to bridge the gaps."

"There's a great deal more to be done, and there are many differences," Haig said after his long meeting with Begin tonight. "But there is also some slight progress."

Haig's earlier visit was described as a "fact-finding" mission to determine whether intensified U.S. diplomatic involvement is warranted and necessary. Haig's discussions today seemed to indicate an affirmative answer and a decision to become more active at a high level in pushing for a renewal of progress.

The secretary of state gave no clear indication in his initial talks today about his preferred mechanism for moving ahead with the suggestions he advanced. There was no talk of a three-way meeting on the foreign ministers' level of Haig, Israeli's Yitzhak Shamir and Egypt's Kamal Hassan Ali, according to Israeli officials.

Haig also did not disclose whether Washington has decided to name a special U.S. negotiator for the autonomy talks, as was done during the Carter administration. There was speculation that Richard Fairbanks, who served as the State Department's congressional relations chief during the first year of the Reagan administration, would play a role in the Middle East negotiations. Fairbanks is accompanying Haig on his present trip and was introduced to Israeli officials as a special assistant.

The planned visit to Washington next week by Egyptian President Hosni Mubarak will present an opportunity to advance the negotiating efforts if the reaction to Haig in Jerusalem and Cairo this week is positive. More important discussions could take place late in February when Mubarak is scheduled to visit Israel.

A U.S. diplomat who is intimately familiar with the U.S.-backed drive for an Arab-Israeli settlement in the past several years said an active U.S. role now is "a necessary effort," whatever its chances of success, to maintain the credibility of the peace process.

While the barriers to an autonomy accord between Israel and Egypt are imposing, the diplomat said, all other roads are much more hazardous.

Israel, fearful that it will lose its bargaining leverage with Egypt after it returns the Sinai, has been pushing for the United States to exert pressure on Cairo for an early agreement. Egypt, on the other hand, is resistant to quick action, arguing that it would be difficult to accept compromises now without appearing to swap them for the Sinai lands.

Haig has bridged this gap, at least rhetorically, by stating, as he did on arrival from Geneva today, that the United States will "do all we can to move the peace process forward, of course without deadlines."