An American arranged agreement on determining the fate of missing victims of the fighting between Greek and Turkish Cypriots failed at the eleventh hour today to win the backing of both sides.

Diplomats still hold out some hope for the accord, under which representatives of the rival groups would sit on a committee with a Red Cross representative to determine the fate of those reported missing in the 15 years of fighting here.

The agreement would be the first significant one between Greek and Turkish Cypriots since the island was effectively partitioned three years ago. The project has become something of a test case of the two sides, ability to agree on anything at all.

The emotionally charged issue will row go to an undoubtedly acrimonious debate this week in the U.N. Human Rights Committee.

Eager to pre-empt the propaganda barrage, American diplomats actively sought to get the consent of both parties to set up the committee to carry out a quiet case-by-case investigation "without assigning blame" and "without seeking propaganda advantage."

So far, neither side has formally accepted the plan, although agreement seemed to be within reach following indications that Turkey had blessed the enterprise. In the end, however, Turkish Cypriots evidently preferred to proceed with the publicity of a U.N. debate.

Turkish Cypriots can get a hearing in the U.N. committee, but they are barred from the General Assembly floor, where only the Cyprus government speaks on the Cyprus problem.

The missing-person issue is an agonizing humanitarian problem for close-knit families on this small Mediterranean island, where several thousand Cypriots have disappeared during the fighting here since Christmas 1963.

Officials here have little hope that any missing persons will turn up alive as a result of investigations now, but the commission could declare people legally dead and convey official findings to bereaved families - to the old people in black, showing fading photographs of sons reported to be alive somewhere. The tragic predicament is familiar to Americans who live through the trauma of those declared missing in Vietnam.

The missing-persons issue is also a political football. By delaying investigation and accepting the U.N. debate, Turkish Cypriots hope to put in the record the cases of people who disappeared in the 1960, presumably abducted and murdered by Greek Cypriot terrorists.

The vast majority of missing persons, however, are Greek Cypriot men and boys, including hundreds, some with American passports, who were captured during the 1975 Turkish invasion but failed to turn up when prisoners of war were released. Some missing Greek Cypriots were murdered by fellow Greek Cypriots during the rightist coup and others went into hiding because they had deserted during the fighting, but there are many cases that point to Turkish atrocitics.

Many diplomats were confident that the U.S.-supported arrangement offered Turkey a way to bury this stigma and also advance its current efforts to show a more flexible posture on Cyprus. It would be helpful, although probably will short of the change needed for the U.S. Congress to lift its arms embargo and economic pressure on Turkey.

As agreement neared and the U.N. debate deadline loomed, the American ambassador here shuttled between the two sides with a swelling attache case full of understandings reached in Washington, New York, Athens, Ankara and the opoosite sides of this divided capital.

Despite their disappointment today, diplomats remained doggedly hopeful over the fact that neither side has closed the door to the plan. These sources believe that "a fair chance" still exists for the committee to be set up after the U.N. debate.