Can the European Community do anything to reverse the growth of its trade gap with Japan?

This question has obsessed policy makers here ever since what they call the "relentless trend" of rising EC-Japanese trade deficits took hold more than a decade ago.

The trade figures for 1985 are unlikely to give these officials any rest: the first nine months of last year showed a 2.3 percent increase in the deficit over 1984, when the total stood at more than $12 billion.

Like the United States, the community complains both about Japan's infiltration of domestic markets and barriers it raises against outside businesses.

But while the EC's trade gap with Japan is second only in size to that of the United States, the community has not been able to win any market-opening agreements with Tokyo of the kind that Washington concluded earlier this month.

In November, the EC took the unusual step of sending three members of its executive commission to Tokyo to put pressure on the Japanese government to set import growth targets and establish a monitoring schedule for its market liberalization program of last July. The commissioners returned to Brussels bearing little more than a promise from the Japanese to ease restrictions on European wine imports.

Such rebuffs have given rise to accusations in the community that Japan treats Europe like a "quaint cultural museum of limited significance" rather than an equal trading partner, one EC official said.

In a reflection of its growing frustration, the community in the past year has stepped up efforts to limit Japanese imports -- imposing anti-dumping duties on hydraulic excavators and electronic typewriters and raising tariffs on videocassette recorders. The community also is considering anti-dumping penalties on Japanese photocopiers, whose manufacturers now have an 85 percent share of the $1 billion EC market.

EC President Jacques Delors will keep up the pressure when he begins a visit to Japan tomorrow.

Beyond repeated pleas to the Japanese and limited retaliatory moves, however, commission officials admit that their range of options is small. The officials are ready to recommend more serious actions, such as restrictions on Japanese banks in the community, but these moves must be made by the member states themselves.

Since there is little agreement among the states on the desirability of protectionist measures, some EC officials doubt that retaliation at that level could be effective.

"Do we have a big stick?" one EC official asked. "I don't think so. We don't really have a position on Japan."

Community efforts to limit imports also are hindered by the changes in the way Japan is doing business in Europe. Japanese manufacturers are rapidly moving their plants here to escape the EC penalties and, at the same time, smooth sources of trade friction.

Japanese Prime Minister Yasuhiro Nakasone, during a visit to France last July, was able to inform his hosts that 28 Japanese firms had begun operations in France in 1984, accounting for a third of the 13,400 jobs created by foreign investments that year. Japanese direct investment in the community rose from $889 million in 1983 to almost $2 billion in 1985.

The benefits from the complex EC-Japanese relationship are another major reason why the community has not taken tougher steps to combat the trade imbalance, EC officials said. European governments, challenged by enduring high rates of unemployment, need the jobs Japanese investments bring. The governments also believe they can learn from watching the fabled Japanese management style at close range.

The outlook for European businesses in their battle with the Japanese is not completely gloomy. The gradual decline of such "natural" Japanese advantages as low wages and an undervalued currency, for example, is expected to boost European competitiveness.

Despite the Japanese market's reputation for discouraging outsiders, a few EC nations actually have developed their own niche in the Japanese economy and built surpluses in certain trade sectors.

Some European companies are making vigorous attempts to compete directly with Japanese producers. Philips, the Dutch electronics firm, plans this year to begin producing 500,000 videocassette recorders annually in a joint venture based in South Korea. Other companies are planning joint ventures with Japanese firms.

Japanese diplomats in Brussels, however, said the Europeans share much of the blame for their inability to penetrate the Japanese market. Kuniaki Asomura, deputy chief of the Japanese mission, said European businessmen lack aggressiveness and are unwilling to shed the "sweat and tears" necessary to master a foreign culture.

EC officials have praised Nakasone as being genuinely committed to easing the West's trade grievances. But they believe he is limited by the tendency for change in the Japanese establishment to flow from the bottom up. "It will be a long time before we have the kind of trade relationship [with the Japanese] that we have with the United States," one official said.