Just over three weeks after the resignation of Socialist Prime Minister Bettino Craxi following a minor parliamentary defeat, there was a glimmer of hope this weekend that the bitter battle for power that has ensued within the government coalition might be on the road to resolution.

The political compromise proposed by Prime Minister-designate Giulio Andreotti, a Christian Democrat, does not envisage Andreotti's forming a new government as he was charged to do by President Francesco Cossiga last week. Instead, according to Socialist and Christian Democratic politicians, the compromise now under discussion would have Craxi, Italy's longest ruling prime minister since World War II, return to the helm of the five-party coalition government that he ran for just under three years.

His renewed mandate as head of the same ruling coalition, however, would be conditioned with a tacit understanding that he would step down sometime next spring and allow a Christian Democrat, almost certainly Andreotti, who has been prime minister five times in the past, to head the government until the end of the current legislative term in 1988.

The government crisis revolves around only one issue: whether Craxi, as the leader of the second largest party in the coalition, should continue heading the government or whether the job should go to a Christian Democrat because it is the largest party.

The dispute has turned into a bitter personal contest between Craxi and the Christian Democratic leader Ciriaco de Mita whose party headed every postwar Italian government until 1981.

The hitch to the compromise being thrashed out this weekend in a series of meetings between Andreotti and other political leaders, including key aides to Craxi, remains that Craxi steadfastly has refused to accept any mandate that would have a time limit.