Prime Minister-designate Bettino Craxi's efforts to put together a new coalition government were threatened today by Republican Party demands for changes in decision-making machinery and revision of Italian foreign policy.

The demands came from Republican Giovanni Spadolini, who as defense minister brought down Craxi's 26-month-old government last week in a dispute over the handling of the Achille Lauro hijacking.

The meeting between the two longtime political rivals -- one of a series held by Craxi with political leaders since he was charged with forming a new government yesterday -- did little to patch up their differences, according to Italian political sources.

Spadolini, who preceded Craxi as prime minister, said that he had stressed the need for greater "collegiality" in decision making. Some Republican sources said that meant he was asking for an inner government executive committee to act on all major decisions.

Spadolini alleged last week that his party, one of five making up Craxi's coalition, had not been consulted adequately on the handling of the hijacking and the later freeing of Mohammed Abbas, a Palestinian leader whom the United States sought to extradite on the ground that he masterminded the hijacking.

A new government also will have to take into consideration "new developments in the Mediterranean and in Italy's European-Atlantic role," Spadolini said, referring to his displeasure over Craxi's pro-Arab foreign policy and his own concern for U.S.-Italian relations.

The Republican Party has only 29 seats in the 630-seat Parliament, but it has influence as a pro-U.S. party with support from business and industry.

Spadolini said he looked with favor on Craxi's decision to go to New York Wednesday to attend President Reagan's summit of western leaders. Republican leaders have made it clear that they fear Craxi's stand on the hijacking threatened Italy's traditional alignment with Washington in world affairs.

Craxi could form a Cabinet without the Republicans, but such a government is not viewed likely to last, and aides to Craxi have said that he is not interested in forming a short-term government.