The Italian government has begun studying ways of providing air cover for its troops in Lebanon as the Italian left increasingly opposes continued participation in the multinational peace-keeping force.

A spokesman said today that Defense Minister Giovanni Spadolini and his joint chiefs had discussed the matter several times this week and had initiated contacts to see if use of the British air base in Cyprus by Italian fighter-bombers was feasible. The Italian troops now in Beirut have transport support facilities at the base. He said that Crete, where there is a U.S. base, also had been mentioned.

Despite a call for withdrawal of the contingent by a Communist leader and other leftists, on Monday the Socialist-led government confirmed its intention to keep the 2,050 Italian troops in Beirut.

Under growing pressure from political groups expressing concern about the likelihood of growing casualties, the Cabinet agreed to study ways to provide protective air cover. So far, one Italian has been killed and more then two dozen wounded.

The spokesman, Gen. Rinaldo Rinaldi, pointed out that because Italy's one aircraft carrier, the Garibaldi, was currently in drydock for repairs, the government was looking for alternative ways to provide air protection. The 4,000-ton destroyer, the Ardito, is to sail Thursday for Beirut. While it could provide some offshore fire cover, military sources said this would not be adequate.

The government's sights are thus set on Cyprus. The agreement of the Cypriot government must be obtained and technical questions about availability of support facilities have to be resolved.

Last week, two leaders of the powerful Communist Party cast doubt on the value of continued participation in the peace force. One of them, Sen. Paolo Bufalini, suggested the troops be brought home.

The Communists--whose votes in Parliament are not strictly needed by the five-party governing majority--have backed the peace force since its inception but say it is no longer carrying out peace-keeping functions.

Citing a "substantially changed situation" and a U.S. decision in favor of a "a direct intervention against the Druze armed forces," the Youth Federation of the Communist Party called today for the withdrawal of the Italian contingent and a "diplomatic initiative" of the Italian government that would lead to U.N. intervention and a cease-fire.

The reference was to recent mediatory moves by Socialist Prime Minister Bettino Craxi, who last week said Druze leader Walid Jumblatt had accepted an invitation to come here for talks. No date has been set. The Socialists would prefer to keep Communist support in the foreign policy area.