Washington — Slowly but surely, the United States is deepening its commitment to Lebanon and to the fledgling government of Amin Gemayel.
To start with, the US is prepared to undertake what one State Department official described as ''a fairly extensive commitment'' to the training and expansion of the Lebanese Army. A US Defense Department team headed by an army brigadier general is now in Lebanon studying the needs of the country's relatively small army.
The administration is also likely to be asked to help to expand the multinational military force which is now supporting the Lebanese government. About 1,200 American marines are participating, alongside French and Italian troops, in this 3,800-man force. Lebanese officials would like to see th multinational force considerably enlarged.
Administration officials have said on several occasions that the marines were expected to be in Lebanon for only a ''limited'' period. President Reagan has said that he thought that the Israeli, Palestinian, and Syrian forces now in Lebanon would be withdrawn in a short period of time, thus permitting the marines and other members of the multinational force to depart rapidly. But the Lebanese themselves are saying that it will take many months to build up their own forces and expand their control to the entire country.
In a breakfast meeting with reporters Thursday, one leading Lebanese Christian leader, former President Camille Chamoun, and his son, Dany, said that they hoped the Lebanese government would request an increase in the multinational force from 3,800 to about 15,000, more than tripling its size.
''The only light at the end of this long tunnel is . . . the physical participation of this administration in helping Lebanon to rebuild itself,'' said Dany Chamoun, who is secretary of Lebanon's National Liberal Party.
The younger Chamoun said that in many parts of Lebanon there is still ''a kind of vendetta mentality'' and that many young men have been ''completely traumatized'' by war, thus necessitating a fairly lengthy transition period to a peacetime mentality. He further said that it might take about two years for the Lebanese Army to assume full responsibility for defense of the country.
A US Defense Department official said, in the meantime, that the Lebanese Army would probably have to be expanded to a size of about 40,000 men from its current strength of less than half that number. The official said that even then it would be more of a territorial and self-defense force than a regular army able to deal, for example, with an invasion from outside the country. The current number of active, full-time soldiers is only about 12,000.
In the interim, the role of the multinational force is likely to be crucial.
State Department officials believe that the Lebanon situation is still highly explosive, in part because Syrian and Israeli forces are now operating virtually face to face in the eastern part of the country. Officials said that with this in mind, Secretary of State George P. Shultz was likely to continue pressing for a rapid withdrawal of the Israeli and Syrian forces when he meets here on Oct. 14 with Israel's foreign minister, Yitzhak Shamir. But the Israelis want to have carefully negotiated guarantees of a Syrian withdrawal as well as of a security zone, some 25 to 30 miles wide, in the south of Lebanon, in order to protect Israel from Lebanon-based attack.
The Americans and Israelis have their differences over how this should be accomplished:
* While the Americans think that a withdrawal of foreign forces from Lebanon can be rapid, the Israelis see no quick way out.
* The Israelis want the forces of renegade Lebanese Major Saad Haddad, whom they support, to play a major role in defending the south of Lebanon, while the Lebanese and American governments do not.
* The Americans want to build up the existing United Nations force now deployed in the south of Lebanon, while the Israelis, based on past experience, mistrust that force.
In the end, a compromise may be found whereby the existing UN force in the south is combined with additional American, Italian and French forces. US Secretary of Defense Caspar Weinberger has said that other nations may be asked to contribute to Lebanon peace-keeping as well.
It was clear in Washington, meanwhile, that tensions between the US and Israel have been eased for the moment. The Israelis seem to realize that they were too harsh in their rejection of the Reagan peace plan for the Middle East and have softened their rhetoric. The Israeli withdrawal from the Beirut airport area and the initiation of a full Israeli inquiry into the recent Palestinian refugee camp massacre have also helped to alleviate tension.