Tension between Moslems and Jews at two sites holy to both religions has underscored sharply how the tenuous coexistence that has prevailed in recent years at their common spiritual shrines can be broken at the slightest hint that the status quo is being altered.

In one case, an unfounded report of illegal construction starting on the Moslem-controlled Temple Mount in East Jerusalem brought an inspection visit by members of Israel's parliament that ended in a scuffle with Arabs protesting the visit. That incident, on Jan. 8, has been followed by another parliamentary visit, an attempt to hold Jewish prayers on the site and an Arab demonstration that was dispersed by police using tear gas and clubs. Most recently, followers of extremist Rabbi Meir Kahane tried to plant an Israeli flag on the site, but they were blocked by police who arrested two of the demonstrators.

In another case, a request by Jewish settlers in the predominantly Arab West Bank city of Hebron to hold unprecedented Friday evening prayers in the main hall of the Mosque of Abraham led to a confrontation last week with local Islamic leaders that ended with Israeli soldiers backing the settlers. Friday is the sabbath for Moslems, who normally pray five times a day in the mosque. The Jewish sabbath begins at sundown Friday.

Mount Moriah, site of the First and Second Temples of Judaism in the 10th and 6th centuries B.C., also has been sacred to Moslems for 1,300 years as Haram Sharif, or the Noble Sanctuary. The mount, with its stunning Dome of the Rock Mosque, is regarded by Moslems as the third-holiest shrine of Islam -- after Mecca and Medina -- and is the place from which Moslems believe the Prophet Mohammed ascended to Heaven.

After the 1967 war, in which Israel captured East Jerusalem, then-defense minister Moshe Dayan ordered the removal of an Israeli flag soldiers had raised over the mount, and ordered control of the area returned to an Islamic committee called the Waqf, or Moslem Trust.

Since then, successive Israeli governments have barred Jews from praying on the Temple Mount. The Waqf was allowed to retain the keys to all of the gates to the mount except for one adjacent to the Western Wall, an arrangement that symbolized Israel's unasserted sovereignty over the holy site.

In 1970, Israel's High Court of Justice rejected a petition by a group of Jewish ultranationalists to pray on the Temple Mount. The attorney general arguing the case for the government asserted then that the mount is within sovereign territory annexed by Israel in 1967, but he noted the Moslems' ancient attachment to the site and the likelihood that Jewish prayers there would offend the Arabs.

The ambiguity of the status of the Temple Mount has led to occasional friction between Moslems and Jews during the past 18 years. For the most part, however, the imposing shrine has remained relatively peaceful under the eyes of the Waqf and Israeli police officers housed in a small substation at the northern end of the broad esplanade.

Deep-seated religious passions over the mount were inflamed throughout the Moslem world on two occasions, once in 1968 when an arsonist set fire to the Al Aqsa Mosque and again on Easter Sunday 1982, when an apparently berserk Israeli soldier shot his way into the Dome of the Rock Mosque with an automatic rifle, killing two Arabs and wounding nine persons.

When members of the parliament's Interior Committee visited the site on Jan. 8 and again on Jan. 14, ostensibly to investigate reports of illegal construction there, many Moslems in the Old City feared that a government-supported effort was under way to change the status quo of Haram Sharif by establishing a precedent of Jewish prayer.

The committee members brought with them Knesset member Geula Cohen, who long has battled for Jewish control of the Temple Mount, and Gershon Solomon, head of a group called the Temple Mount Faithful, whose members frequently have been arrested while trying to pray at the site. The demonstrations and counterdemonstrations followed.

Faisal Husseini, a member of the Waqf, said, "We all know Gershon Solomon. We also remember Geula Cohen and her visit to Hebron, which marked the beginnings of Jewish settlement in the heart of the town . . . . They the Moslem worshipers were afraid that this was not just a visit of Knesset members, but that they would stay there or take some further action. Just imagine that a group of Moslems goes to pray near the Wailing Wall. Can you imagine the reaction?"

In the aftermath of these events, the comments of Cabinet ministers have reflected the strains in the government coalition.

"To turn a religious struggle, or even a military struggle, into a religious, Islamic-Jewish struggle would be the greatest mistake we could make," said Labor Party Defense Minister Yitzhak Rabin, warning of "disastrous" consequences unless the Temple Mount controversy is eased.

But Deputy Prime Minister David Levy, a leader of the coalition's Likud wing, declared in a speech in Hebron, "If in these days, a Jew who wraps himself in a prayer shawl on the Temple Mount is called a provocateur, then we are all provocateurs."

In Hebron, where Arab resentment over Jewish settlers moving into the heart of the city since 1979 has brought increasing friction, the latest controversy focuses on the area of the mosque, sacred to Jews and Christians as the burial place of the biblical patriarchs Abraham, Isaac and Jacob. The shrine is called Machpelah Cave by Jews and Haram Khalil by Moslems.

From the time Israel occupied the West Bank in 1967 until last week, Jewish prayer has been restricted mostly to the outside of the mosque or an area between the cenotaphs of Abraham and his wife, Sarah.

A spokeswoman for the West Bank military government described the new move as a "temporary change of policy" requested by Jewish settlers.

The settlers "asked to be allowed to pray inside because of the winter rain, so it was decided to let Jews and Moslems share Isaac's room until March 31. It's quite a sensitive subject because of the question of the status quo," said the spokeswoman, adding that the decision was made at a political level higher than the West Bank military governor.

However, Mustafa Natche, the deposed Arab mayor of Hebron and the leading Palestinian figure in the city, charged the it was part of a "step-by-step campaign to force Moslems out of the Mosque of Abraham."

Natche said that when the head of the Waqf, or Moslem trust of the mosque, Sheik Azzam Hatib, rejected orders by the military governor to restrict the Moslems' prayer area, Israeli troops on Friday night cordoned off the building and ordered Arab employes to remove prayer rugs from most of the main hall.

During Arab demonstrations Friday, Palestinian sources said, an Arab attacked an Israeli soldier with a knife. West Bank military government officials said that the incident had been exaggerated and that a "demented" man had been disarmed quickly by other Arabs before getting near the soldier.

This week, the Army troops erected six-foot-high steel barricades in the hall housing the cenotaphs of Isaac and Rebecca, leaving Moslems a small, segregated area behind the tombs for their prayers. An Army officer said the portable barricades were for Friday evening Jewish prayers.