Pope Paul spent 40 minutes today discussing the problems of Rome with the city's Communist-backed mayor, Giulio Carlo Argan, in a private audience heralded by the press as a significant step forward for the powerful Communists.

The Vatican, however, apparently trying to counter rightist Catholic opinion that the church is over-friendly with Communist leaders, played the meeting down.

The Vatican newspaper L'Osservatore Romano, in contrast to its treatment of the Pope's meetings with past Christian Democratic mayors of Rome, did not mention the private audience. It carried a brief papal audience with the mayor together with all the city councilmen.

The Pope's meeting with Argan and 14 other City Council members was the first since a leftist coalition won control of the city government last August despite the Vatican's warning to voters that a Communist victory would turn Rome into a "Godless city."

The Pope and Argan had met informally twice before today's audience but only briefly - at a public ceremony Dec. 8 and at a New Year's Mass.

Political observers suggested that the meeting - described as harmonious - presage a greater accomodation between the Communist Party and the Vatican in city and Italian affairs.

The Pope and mayor traditionally exchange New Year greetings each year, but this year's meeting took on unusual political significance because of the leftist victory.

"We hope that the singular character of Rome might always be taken into account," said the 79-year-old pontiff, who is also bishop of Rome. "It is this symbiosis of sacred and profance life which distinguishes the events of Rome through the centuries . . . which can help to take the necessary measures aimed at improving the welfare of its inhabitants, especially those in need-in charity."

Argan, a 57-year-old history professor and political neophyte, told the Pope and City Council was burdened by massive indebtedness run up during 30 years of rule by the Vatician-back Christian Democrats. He ticked off the ills besetting Rome, especially "the greedy building and land speculation" that he said caused a scarcity of jobs and housing.

The Italian magazine L'Europeo reported last week that the Vatican and its religious orders own at least one-fourth of the real estate in the Italian capital and that they seem to have began divesting themselves of old religious buildings at windfall profits and without paying taxes.

In a radio interview after his Vatican audience, however, Argan said it would be giving a "false outlook" to hold the religious community responsible for land speculation in Rome.

The 1929 Concordat, the treaty governing relations between the Holy See and the Italian government, grants extraterritorial immunity for some church property in Rome. Parliament has recently given approval for continued talks between the government and the Vatican to update the accord.