As a fierce May sun scours Tel Aviv's Bauhaus-style "White City", it's cool down in the compact "cellar" theatre of Habima.

One of four auditoriums in the sleek, streamlined headquarters where Israel's national theatre has returned after the building's five-year restoration, it has seating on three sides and eyeball-to-eyeball contact between actors and audience. Ilan Ronen, Habima's artistic director, learned his trade with Mike Alfreds at the Khan Theatre in Jerusalem before running Tel Aviv's other leading company, the Cameri. He chose this intimate basement as the best preparation for Habima's two performances in Hebrew of The Merchant of Venice, part of the Globe's multi-lingual Shakespeare festival.

The actors run through a few scenes for me, including Shylock's "Hath not a Jew eyes?" speech. Bruised, plaintive, bewildered more than furious, Yaakov Cohen makes a heart-rending Shylock. A Sephardic Jew who migrated from Morocco, Cohen proved a divisive choice for an always-contentious part. "It's the first time in Israel that a Sephardic actor has played this role," says Ilan Ronen, who thinks that prejudice against Jews of Middle Eastern origin persists. When people criticised the casting, "the official reason was that he's a comedian. The unofficial reason is that he's a Sephardi."

Cohen (his words translated by his colleagues) explains his stance on Shylock. "He's not like this," he says of the urge for revenge, "but he's brought to the edge. He's a victim of all that people are doing to him." When the pound-of-flesh moment arrives, and an imaginary revenge on his Christian tormentors takes the form of an all-too-real knife, "He wants to do it, yet he wants someone to stop him. Inside, there's a big turmoil."

As Ronen notes, Cohen's Shylock "creates a lot of empathy. There's no aggression coming from him. You feel that he's victim rather than a villain. He's a result of the circumstances he's been living in. Politically, when you push someone to the edge, he can be very violent. This is what happens to minorities." A little later, Ronen offers a comparison that so startles me that I ask him to repeat it. For the director, Shylock as his vengeance nears becomes "almost like a shahid". A shahid, a martyr? The Arabic word often implies a suicide bomber. So – this is my gloss, not explicitly Ronen's – the slow-kindling anger of the archetypal Jewish scapegoat illuminates the nothing-to-lose rage of Islamist militants. "And if you wrong us, do we not revenge? If we are like you in the rest, we will resemble you in that."

Habima owes its origins as an official theatre to a certain Joseph Stalin. After the Russian Revolution, the then Commissar of Nationalities approved the quixotic venture of a group of Jewish actors – led by Nachum Zemach – who had begun to perform in modern Hebrew. With support from the legendary Stanislavski, they flourished as a co-operative, and toured abroad to acclaim – especially for their signature show, Ansky's The Dybbuk. After Soviet anti-Zionists gained the upper hand, Habima migrated: first to the US then, in 1931, to Tel Aviv in British-mandate Palestine. In the foyer of the gleaming new HQ, a slab rescued from the old theatre records in Hebrew the High Commissioner's inauguration of their first home.

In Britain today, Habima's visit to the Globe has come to mean only one thing. To boycott or not to boycott? That has been the sole permitted question. Actors, writers and directors – from Mike Leigh to Mark Rylance – called for Habima to be disinvited.

The company receives around 30 per cent of its funding from the Israeli state. It has on a handful of occasions performed in Israeli settlements on occupied land in the West Bank – notably in fast-expanding Ariel. The UN deems these settlements illegal under the Fourth Geneva Convention of 1949. In the UK, the theatre community has split over the boycott campaign, and the Globe shows will not pass without incident. "With music, you can continue to play," Ronen tells me, a little wearily, as he recalls the disruption during the Israel Philharmonic's Prom last summer. "In theatre, I'm not sure that you can do the same. So people might get two shows on one ticket! I hope that we'll survive."

Some Habima supporters, he stresses, "were against the whole idea of taking what they thought was an anti-Semitic play to London." As for the settlement issue, home-grown criticism started earlier and hit harder than anything yet heard beside the Thames. In 2010, 60 Israeli theatre professionals signed an open letter condemning publicly funded companies for putting on shows in Ariel. Earlier this month, dramatist Vardit Shalfy censured Habima in the liberal daily Haaretz for travelling down an "apartheid highway" "to perform at a glamorous, brand-new Hall of Culture...within touching distance of a refugee camp whose residents live under Israeli military rule".

For Ronen, Habima must perform for all Israelis. To objectors, the settlements are not Israel. Ronen notes that his company has pioneered productions that aim to cross the region's abysmally deep divides. "From the first moment I came here, I knew that I wanted to use my personal connections... to create dialogue with Palestinian artists." Stories of migration, of exile, of displacement, fill the recent repertoire. The Third Generation, co-produced with Berlin's Schaubühne, brings together Palestinian, Jewish and German perspectives to interrogate the legacy of the Holocaust. A Railway to Damascus, which Habima took to Ariel, explores the roots of today's tensions in the upheavals of the 1940s. "I think it's very important that people sitting in Ariel should understand the sources of the Arab-Israeli conflict during the British Mandate," Ronen insists. Theatre can help to puncture the political bubble of life in the settlements: "I should confront the people who are against my ideology." He adds that "If we were censored, then you would have a case for protest – and I would protest with you."

Ronen argues that the motive for each boycott shifts: collaboration with the army in the case of the Israel Phil; Ariel for Habima. Behind them, he suspects, lies the drive to exclude all Israeli artists from the international arena. When I attended a post-performance discussion at the Globe after the Palestinian actors of Ashtar Theatre had given their excellent Richard II, their objections to Habima's invitation stemmed from the general platform of the BDS (boycott, disinvestment, sanctions) campaign. It did not pivot on the settlements. "It's not just Habima, it's any Israeli organisation," said Nicola Zreineh of Ashtar, who played the rebel Bolingbroke. "For us as Palestinians, we call for a boycott for as long as we are under military occupation; for as long as there is no justice in our country; for as long as we are deprived of very basic rights as human beings.

"It's possible for Bolingbroke to enter Jerusalem," he said – at the play's end, the newly crowned Henry IV plans a cleansing pilgrimage to the Holy land – "but not for me! Don't forget that we are besieged – all of us, all the Palestinians."

Habima and Ashtar seem to stand on either side of a partition as high as the Separation Wall that snakes like a vast scar across the lovely springtime hillsides. As for Ilan Ronen, he voices his support for the "fantastic" performing arts of the Palestinian West Bank. And he adds, enigmatically, that "even if there is no official dialogue, there is always mediation. There are meetings all the time." On the boycott call, however, Habima stands its ground. "We come to the Globe along with 37 countries and languages. And this is the only theatre, and the only language, that should be boycotted? Everything is OK in those other countries – no problem at all? Artists should not boycott other artists... I think, as an artist, that this is wrong. We should have a dialogue with everybody. We should discuss and disagree."