The United States is slamming the door shut on imports of three categories of specialty steel until mid-October because foreign suppliers speeded up shipments after President Reagan imposed world-wide quotas on July 5.

Trade officials said foreign suppliers stepped up their shipments in an effort to get as big a share of the U.S. market as possible before the United States set limits for individual countries.

As a result, the quotas set for the first 90 days of imports already have been filled for one category of specialty steel and are close to being filled in the other two categories, U.S. Customs Service officials reported yesterday. The quotas took effect two weeks after they were imposed and the first 90-day period does not end until Oct. 20.

Industry executives said shutting down imports for the next six weeks is unlikely to help domestic sales, since there are large inventories of foreign-made specialty steel products still on hand.

But the abrupt ban on imports is expected by trade officials to force overseas suppliers into quick negotiations for orderly marketing agreements that will guarantee them a fixed share of America's specialty steel market.

Negotiations are scheduled to start next week with Canada, Sweden and Austria, who have indicated an interest in setting these marketing agreements, and talks with the Japanese are expected to begin when U.S. trade officials visit Tokyo in advance of President Reagan's trip there in November.

The European Economic Community, however, has decided to go another route by demanding compensation for losses suffered because of United States restrictions on steel imports from its companies. This strategy is seen here as a reflection of the inability of the 10 EEC countries to agree on how to share the available business in the U.S. market. It carries the danger, however, that the Common Market will end up sharing a small global basket with other nations that failed to negotiate orderly marketing agreements with the U.S.

The president ordered quotas on specialty steel imports for the next four years because of what he called "the pervasive nature of unfair trading practices around the world."

In addition to imposing quotas on three kinds of specialty steel, the president placed tariffs on other kinds of low-volume, high-price alloys; those steel products are not affected by the holdback on imports.

Customs officer William J. Wagner said imports of stainless steel wire rods reached their 90-day limit at 11:35 a.m. Tuesday. All shipments received since then are either being turned away or placed in bonded warehouses until the next 90-day period starts Oct. 20.

As of Thursday, 91.6 percent of the 6,750 ton 90-day quota for imports of steel bars and 86.8 percent of the 5,600-ton, three-month quota for alloy tool steel had been shipped into this country.