If David Cameron is feeling a tad frustrated by the lack of progress in the Middle East – breaking with usual diplomatese during a visit to Turkey todayto brand Gaza a "prison camp" – then he is not the only one. "Everything is stuck," sighs Jamal Zahalka, a Palestinian member of the Israeli parliament on a visit to London. The small Arab nationalist party he leads is formally committed to the two-state solution which would see a Palestinian state alongside Israel, but he sees no prospect of it. Those in charge are interested only in "conflict management, not resolution", he says. Talk of two states lives on in the seminar room, but it is not on any horizon visible in the real world. I'm told the Palestinian president, Mahmoud Abbas, is "on the verge" of giving up on the two-state approach he has believed in since the 70s. And there are plenty of Israeli Jews who share that pessimism.

So it's hardly surprising that people are searching for other answers, even lighting upon an idea long confined to the margins, advocated only by extremists and dreamers: the "one-state solution." This holds that the two warring peoples should not live in two separate entities, with separate flags and governments, but as citizens of a single state. Jews and Arabs would no longer face each other anywhere as occupier and occupied, as they do now. They would live side by side, in a shared country.

But here's the twist. This vision comes not from diehard Arab rejectionists, who refuse to countenance any arrangement which allows for a distinct Jewish state, nor from old-school student lefties from the 80s, once tireless in their advocacy of a "sec dem" – as in secular, democratic – state. No, the new advocates of the one-state solution are the Israeli nationalist right, including several luminaries of the settlers' movement.

The most prominent to break cover was Moshe Arens, who served as both foreign and defence minister in the Begin-Shamir era two decades ago. He wrote last month that it was time for Israel to look at "another option", one that would see a single state on the land that is now Israel and the West Bank. The Palestinians who live there would, wrote Arens, no longer be a people under occupation but full citizens. As Haaretz has reported, its editorial jaw dropping, the former minister has been joined by an array of rightist eminences, including a founder of the settlers' organisation, Gush Emunim, a former chief of staff to Binyamin Netanyahu, a senior Likud member of the Knesset, and the parliament's current speaker.

To understand quite how taboo-busting this is, recall the case of the scholar Tony Judt. In 2003 he too floated the notion of a single state to be shared by both Arabs and Jews. Judt's essay was instantly denounced as anti-Zionist heresy; among many Israel supporters he was rapidly ostracised. Thereafter Judt was identified as firmly on the radical left of the Israel-Palestine debate. Yet now those same thoughts are aired on the radical right.

For all that, it is not so hard to see why this once forbidden notion now appeals to Israel's nationalist camp. The two-state solution may be conventional wisdom across the globe, endorsed by almost all democratic governments, but for settlers and their allies the very idea reeks of trauma: any division of the land is assumed to entail the dismantling of the towns and villages they call home. For the devout, this means leaving places they regard as part of the ancient biblical homeland. Some threaten armed resistance; rightwing soldiers warn they will refuse any order to evacuate settlements. But if there's a single state, all that trauma can be avoided. "It's preferable for the Palestinians to become citizens of the state than for us to divide the country," says Knesset speaker and Likud MK Reuben Rivlin. What's more, there would be no place for the current wall, or separation barrier, that some rightists believe disfigures and artificially divides what should be the sacred, and whole, Land of Israel.

How has the right come to this new realisation? In a word: Gaza. Before Israel's 2005 disengagement from the strip, any talk of a single state struck demographic fear into Israeli hearts. For if Israel were to absorb the Palestinians of both the West Bank and Gaza, their combined number would instantly endanger the country's Jewish majority: there would soon be numerical parity between Jews and Arabs. But for Arens et al, that link to Gaza has now been severed, taking its 1.5 million Palestinians off the books, as it were. That leaves the Palestinians of the West Bank, estimated variously as 1.5 or 2 million, whom Israel could just about absorb.

At first glance, this new direction might look appealing to Palestinians too. Plenty are sick of waiting for a state that never materialises: why not continue the struggle on a new front, using their strength of numbers at the Israeli ballot box? There would be immediate benefits. Families split by the 1967 battlelines might be unified once more; Palestinians would have access to jobs and economic opportunities inside Israel.

But they should pause. For one thing, the rightist one-staters are making no instant promises: the granting of citizenship would, they warn, be "gradual". Nor are they offering a truly binational state that would grant equal status to the two nations. Instead Israel would remain a Jewish state, with West Bank Palestinians offered only the civil rights owed to them as individuals, not any national, collective recognition. They would be treated much the same way as the 1 million Palestinians who are already citizens of Israel. Given the long history of discrimination that community has endured, that might not be such an enticing prospect.

When I put the idea to Zahalka's fellow parliamentarian, Haneen Zoabi, she was adamant: to accept such an offer would amount to surrendering the Palestinian claim of sovereignty over the West Bank. "We will never give that up," she said.

Israeli Jews have every reason to be equally sceptical of this one-state talk. If Zoabi is right that West Bankers will not accept citizenship in a Jewish state, then the alternative is a properly binational entity. Yet what evidence is there that two peoples who couldn't get along well enough to negotiate a divorce will do better in a marriage? Support for such an idea is close to zero. It is hard to see that changing so long as Israelis, and Jews around the world, continue to yearn for the thing that almost every other nation takes for granted: a state of their own.

And yet this development is not to be dismissed out of hand. The one-staters of the right say they are raising the issue now because the status quo has become intolerable to world opinion, corroding Israel's legitimacy. "The international community takes that stance because we are still occupiers," one Likud MK said. "There will be greater legitimacy when the occupation ends."

That the right has finally reached this realisation is good news in itself. Avraham Burg, a former Labour politician on an ideological journey of his own, says these new noises from the right will put pressure on Netanyahu, forcing him to reach a two-state solution before it's too late. "The days of the two-state solution are numbered," Burg says. "It's not there forever, with no expiration date."

The Israeli right are banking on the assumption that it's already too late. It's up to those who still believe that two states represent the last, best hope for both Israelis and Palestinians to prove them wrong.

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