Jerusalem — Israel is carefully watching the political evolution of Bashir Gemayel from warlord to president.
The Israelis, who have been secretly supplying arms and training to Mr. Gemayel's Christian Phalangist forces for years, had been clearly hoping for his election. Government sources have hinted at a promise from Gemayel to eventually establish formal diplomatic relations with Israel, and Prime Minister Menachem Begin has predicted an Israeli-Lebanese peace treaty ''in the near future.''
Mr. Gemayel is also seen as the Lebanese political figure most likely to prevent armed Palestinians from returning to the border with Israel.
But there is a growing sense in Israel that Gemayel, who was once the embattled warlord in need of outside support, has different interests now that he is the leader of a nation - particularly a nation that wants to remain part of the Arab world.
Israeli officials have noticed a clear distancing from Israel by Gemayel, a distancing that dates to when Israel launched its incursion into Lebanon in June. Phalangist leaders had for years been urging Israel to do just that, in order to clear the country of the Palestine Liberation Organization. According to Israeli sources, the Phalangists had pledged to undertake the job of clearing west Beirut of the PLO if the Israelis reached the city.
When it came time, however, Gemayel's forces kept clear of the conflict once the Israelis crossed the border. There was an embarrassed pause when the Israelis reached Beirut and waited for the Phalangists to move against their long-time PLO foe bottled up in the city. But the Phalangists, in an ''after you , Alphonse'' routine, indicated clearly that they would prefer that Israel do the job.
The Israeli war machine could do it better, they argued. But the Phalangists also wanted to preserve their trained manpower for the internal Lebanese struggles that might come afterward.
The Israelis undertook the job of prying the PLO and Syrians out of west Beirut, relying on air strikes and artillery.
Political observers in Jerusalem say the main reason Gemayel refused to commit his forces in west Beirut was political rather than military. It was a reluctance to appear too closely linked to Israel and become a party to the invasion.
With elections for the presidency looming, he was clearly concerned about Lebanon's good standing in the Arab world, on which it is economically dependent. He also evidently did not want to appear an Israeli puppet to his own people.
Israel's reaction to Gemayel's suddenly low profile toward them has been a mixture of annoyance and understanding. Apparently piqued at Gemayel's refusal to go into west Beirut, Defense Minister Ariel Sharon extended the territorial sway of Israel's other Christian ally in Lebanon, Maj. Saad Haddad, commander of the strip immediately north of Israel's border.
With the expulsion of the PLO from south Lebanon, the Israelis did not invite Gemayel to fill the vacuum but extended Haddad's domain as far north as Sidon.
Nevertheless, most Israelis accept that Gemayel's position will require him to maneuver tactfully. They take solace in the thought that while Realpolitik will prod Gemayel toward bridge-building with the Arab world, he is unlikely to cast off the Israelis, who are his ultimate buttress against the instability of that world.