The State Department said yesterday that the United States has asked the Soviet Union to open negotiations on putting outer space off limits to such hostile as satellites that destroy each other.

Although the department did not disclose how the request was made, now when, administration sources said yesterday it was made last month to Soviet officials in Washington. U.S. officials suggested that the first meeting he held sometime in April, sources said.

The Soviet Union has not yet replied to the request, according to aministration officials.

Defense Secretary Harold Brown, administration sources said, felt that a meeting next month wouldn't allow enough time to corrdinate the U.S. government's effort and unsuccessfully argued for a later date.

Brown sources stressed, welcomes negotiations to ban hostile space activities, and did not argue for waiting until the United States had such weapons in hand. The Soviet Union has a satellite killer system while the United States is still developing one.

The Soviet antisatellte system requires that the killer satellite track and intercept the enemy satellite and then explode near it. The United States is working on several faster-reacting systems.

The Vought Corp. of Dallas, for example, has a contract with the Air Force to build a flying test model of a tiny vehicle that would leave the mother rocket once in space and home in on the heat of the enemy satellite. The tiny satellite killer would crash into the enemy satellite while flying thousands of miles an hour, relying on the force of collision rather than explosives for destruction. This heat-seeking satellite killer is expected to be ready for flight testing by 1980.

President Carter and Defense Secretary Brown have said that the United States is hopeful of negotiating a limitation on military activity in space. Said State Department officials yesterday in confirming that the United States had asked the Soviets to begin talk: "We don't want to see the arms competition extended into outer space."

The U.S. military has become increasingly dependent on space satellites for reconnaissance, communication, navigation and monitoring such Soviet activity as rocket launchings. Brown said in his annual posture statement that the Soviet "with their present capability" in space "are leaving us with little choice" but to respond to those efforts both defensively and offensively.

"Because of our growing dependence on space system," Brown said, "we can hardly permit them to have a dominant position in the ASAT [antistellite] realm."

Brown said last October that the Soviets had put into operation an antisatellite system. The Pentagon's fiscal 1979 budget called for sharp increases in military space programs, a trend that would become more pronounced in future years if a broad outer space treaty covering antisatellite system is not negotiated.