A Swedish policeman may have seen William Bradford Bishop Jr., the missing State Department official wanted for the murders of five members of his family, on a Stockholm street shortly after a friend of Bishop's reported seeing him twice in a downtown Stockholm park.

But because Swedish police did not get a picture of Bishop froim the FBI until about a month after the alleged July sightings, the officer could not positively identify Bishop. Although the policeman was aware that Bishop was being sought in the city, he had only a written physical description to go on.

A spokesman for the U.S. Embassy in Stockholm said yesterday Bishop's picture was turned over to Swedish police sometime last August. The woman reported seeing a bearded man resembling Bishop last July 1 and 4.

The possible third sighting of Bishop in Sweden was reported to the chief of Stockholm's criminal investigation division when he distributed Bishop's picture to his detectives and one of them said he may have seen the man earlier according to Ben Schaumburg, the security officer at the American Embassy in Stockholm.

While there has been a flurry of excitement this week, prompted by an FBI request that Swedish authorities enlist the aid of Stockholm residents in looking for Bishop, police there say the trail has been cold since last summer.

"We did everything to find him" from the July report until about September, said Inspector Gunnar Larsson of the Interpol office of the Swedish police. But when that investigation turner up no further clues, "we have done nothing much" since, until this week's request from the FBI, he said.

Larsson said yesterday that the woman who reported seeing Bishop told Swedish police she had known Bishop for 14 years. She worked in the Swedish Embassy in Addis Ababa, Ethiopia, in the mid-1960s when Bishop was assigned to the U.S. Embassy there. She said she was a frequent visitor in the Bishop home and described herself as the "best friend" of Bishop's wife, Anette.

The woman's identity has been kept secret because she feared Bishop as she knew she could be "a very important witness" in the worldwide search for him, Larsson said. He added that the woman is in her 40s or 50s, and was married to an Ethiopian at the time she lived in Addis Ababa. She has since returned to Stockholm, dropped out of the Swedish foreign ministry and was employed until recently by a private firm in Stockholm, Larsson said.

Larsson said the woman first came to Swedish police about Bishop in November 19778 saying she thought Bishop might attempt to take refuge in Sweden "because he knew so many Swedes." The woman said she came to police after she and other Swedish friends who were mutual friends of the Bishops "were discussing this terrible act and speculatedon where Bishop might have gone if the were alive.

"She said they agree that Sweden and Australia were among places he might try to hide," Larsson recalled.

Larsson said he sent a man to the woman's apartment to interview her, but "there wasn't much we could do about [her fears.]". Larsson said Swedish police "already were aware that Bishop was wanted, there was an internationa warrant out for him."

The Interpol inspector is somewhat skeptical of the woman's report, noting that she had not seen Bishop for about 10 years, and that the man she saw was wearing a beard.

Larsson mused yesterday that "the best theory in this case is that Bishop committed suicide and was eaten by bears in the Smoky Mountains."

Bishop's station wagon was found abandoned in a campground of the park on the North Carolina-Tennessee border on March 18, 1979, 17 days after his wife, mother and three sons were besten to death in their suburban Bethesda home.

A massive search of the huge park, by foot and air, failed to produce one trace of Bishop. Police later learned that Bishop, who was an experienced camper, had taken private lessons in white-water canoeing sometime before the killings. One theory is that Bishop escaped from the mountainous park by canoe.

The last confirmed sighting of Bishop occurred March 2, 1976, in Jacksonville, N.C., when the then 39-year-old foreign service officer used his Bank Americard to biy $15.50 worth of supplies in a sporting goods store.

The battered bodies of Bishop's wife; mother, Lobelia, 68; and sons Bard III, 14, Brent, 10, and Geoff, 5, were recovered earlier that day about 100 miles north of Jacksonville when a forest ranger found their burning bodies in a shallow grave.