The site in question is Joseph's Tomb, which is revered by Jews as the burial site of the Old Testament figure, son of the patriarch Jacob, who was sold into slavery by his brothers but rose to become viceroy of Egypt.
The tomb's location - inside a compound in the Palestinian refugee camp of Balata in the northern city of Nablus - means the faithful can visit only once a month, under tight military security, and only at night.
But Israel's influential national-religious community, on which Prime Minister Benjamin Netanyahu's coalition government leans heavily for support, says the night-time visits are demeaning and a surrender of religious rights.
"It is ridiculous that Jewish pilgrims can only come like thieves in the night, under military protection, to visit Joseph's Tomb which is a sacred place for us," they say.
And now many are demanding that Israel be given complete control over the site.
"Joseph's Tomb should be under Israeli sovereignty," cabinet minister Yuli Edelstein told AFP on one late-night pilgrimage, among hordes of worshippers pressing into the whitewashed interior of the tomb.
"It's very unfortunate that Jews have to visit this holy place only at night because of the security problems," he told AFP, although he ruled out any chance of Israel seizing the tomb by force.
Once a month, a convoy of around 20 coaches, each carrying 50 people, turn up in the dead of night so they can pay their respects at the stone tomb which was renovated a year ago by the army.
Around 600 soldiers are mobilized for the operation, which also requires close coordination with the Palestinian security forces.
Although the visits are monthly, pilgrims have to apply "five or six months" in advance for a place on the convoy, complains Rabbi Nissim Attias who lives in the nearby Jewish settlement of Elon Moreh.
The spot has long been a destination for Jewish pilgrims but it has also been a place of bloodshed.
It was renovated after Israel seized the West Bank from Jordan in the 1967 Six-Day War and was reopened to Jewish visitors in 1995, two years after the 1993 Oslo Accords which saw Israeli forces pulling out of Nablus.
'Every house has a key'
But in October 2000, at the start of the second intifada, or uprising, Palestinians attacked the site, vandalizing the tomb and partly destroying it after driving out an Israeli border police detachment stationed there, one of whom died in the fighting, along with six Palestinians.
Seven years later, after the intifada petered out, the army began allowing pilgrims to join monthly convoys to the site, but limiting the visits to the night hours in order to avoid all contact with the local Palestinian population.
For the Palestinian Authority, which has controlled the area since 1995, the question of handing the site over to the Israelis is out of the question.
"We reject this demand because this is a part of Palestinian land," said Mahmud Habbash, the Palestinian minister for religious affairs.
"Until now, it hasn't been confirmed as a Jewish site," he said, pointing to studies showing the site was a burial place for a Muslim sheikh.
"Even if we accept it as a Jewish site, we will not accept any Israeli control here or on any of our land."
For the Jewish faithful gathered at the tomb, the site has a deeper political resonance, with Joseph hailed as "the first Zionist" - the one who held firmly onto his faith despite being surrounded by non-Jews, in reference to his stay in Egypt, says Gershon Mesika, a settler leader in the northern West Bank.
"It is time for the Jewish people to stop being humiliated and to embrace Joseph in broad daylight," said Elyakim Levanon, a leading settler rabbi.
"Every house has a key and without a key, you cannot come in. The key to enter the Land of Israel is Joseph," he told AFP.
"Joseph is the symbol of the rebirth of the Jewish people in their own land," declared veteran settler leader Benny Katzover.
"That is why the Arabs do not want us to have access to this symbol."
Meanwhile, the worshippers make the most of their monthly pilgrimage as they jostle around three stone pillars in the small circular room.
"May God grant our prayers for the sake of Joseph the wise, of Efraim and Menashe," they intone.
Black-clad ultra-Orthodox, Jewish settlers, clean-shaven soldiers and elderly men with long grey beards, they all come to worship, to dance, to sing.
They kiss the stone draped in black cloth, they prostrate themselves on it, swaying, mumbling and praying, repeating the name "Joseph" over and over again.
And all around them, blissfully unaware of the religious ecstasies filling this sacred cave, the great city of Nablus sleeps.