Last year, he suggested publicly that Egypt’s president “go to hell.” In the Israeli parliamentary elections, to be held Tuesday, he is running on a vow to require Arab citizens to sign a loyalty oath. As his campaign slogan asserts with a sly wink at Jewish voters, Avigdor Lieberman “knows how to speak Arabic.”

Mr. Lieberman does not know Arabic and will not, by all polls and predictions, become the next prime minister. But his popularity has been climbing so steeply that his party is now expected to come in third, making him a likely power broker with an explosive and apparently resonant political message: Israel is at risk not only from outside but also from its own Arab population.

“It no longer matters whether Lieberman will get 19 seats, as some polls indicate, or merely 15,” noted the political commentator Sima Kadmon in Friday’s Yediot Aharonot newspaper. “He is the story of this election campaign.”

The front-runner and likely prime minister remains Benjamin Netanyahu of the conservative Likud Party. Close behind him is Tzipi Livni, the foreign minister and leader of the centrist Kadima Party. Until recently, Defense Minister Ehud Barak, leader of the left-of-center Labor Party, was in third, having been bolstered by Israel’s recent war in Gaza.

Now Mr. Lieberman’s Yisrael Beitenu (Israel Is Our Home) holds that slot. He and his party, who have drawn support away from Likud, may well be part of an eventual coalition government or lead the opposition.

Most of the political establishment, including members of the Likud, Kadima and Labor Parties, are furious and afraid of either possibility because they broadly consider Mr. Lieberman a demagogue. They fear that his focus on a normally submerged paradox of political life here — how a state made up of Jews and Arabs can define itself as both Jewish and democratic — undermines a delicate coexistence. They say he is drawing in Israeli Jews who feel the country needs a greater display of power to survive.

“I am afraid of this guy, and I dislike him,” said Shmuel Sandler, dean of social sciences at Bar Ilan University, an institution that emphasizes Jewish identity and values. “He appeals to simple-minded voters. Average Israelis feel that we have given up territory, and at the same time the Arabs don’t want to accept the Jewish nature of the state.”

Israel’s military assault on the Hamas rulers of Gaza has helped Mr. Lieberman in two ways. First, he presents himself as a strongman eager to confront Israel’s enemies. At the same time, Israeli Arabs sympathetic to Gaza protested the war, which incensed many Jews.

“The biggest boost his campaign had were pictures of Israeli Arabs waving Hamas flags during the Gaza war and shouting ‘Death to the Jews,’ ” noted Abe Selig, a reporter for The Jerusalem Post who has been covering Mr. Lieberman.

But he is not classically right wing — he is less doctrinaire about land and is not religious — and his iconoclasm seems to be drawing voters from surprisingly diverse political tendencies.

An immigrant from the Soviet Union — he was born in Moldova and moved here in the late 1970s — Mr. Lieberman, 50, wants to ease the paths of those of his fellow immigrants who are children of mixed marriage and looked down upon by the rabbinate.

Unlike many on the far right, he favors a two-state solution with the Palestinians. He wants to trade away parts of Israel that are heavily Arab to the future Palestinian state in exchange for close-in Jewish settlement blocs in the West Bank. He lives in such a settlement near Bethlehem.

His loyalty oath would require all Israelis to vow allegiance to Israel as a Jewish, democratic state, to accept its symbols, flag and anthem, and to commit to military service or some alternative service. Those who declined to sign such a pledge would be permitted to live here as residents but not as voting citizens.

Currently Israeli Arabs, who constitute 15 percent to 20 percent of the population, are excused from national service. Many would like to shift Israel’s identify from that of a Jewish state to one that is defined by all its citizens, arguing that only then would they feel fully equal.

Mr. Lieberman says that there is no room for such a move and that those who fail to grasp the centrality of Jewish identity to Israel have no real place in it.

Oddly, Mr. Lieberman and those who support him often say that the loyalty oath mirrors an American practice, apparently mistaking the naturalization process for a universal requirement for all United States citizens. For example, Uzi Landau, the party’s No. 2, said recently, “In the United States, whoever wants to be a citizen has to pledge allegiance to the country and its Constitution, know the anthem, be familiar with the flag and its history.”

Taken together, Mr. Lieberman’s proposals aim toward an ethnically purer Jewish state, in many ways a classically conservative goal. But his willingness to give up land where Israel is narrowest, and around Jerusalem, for the sake of reducing the Arab population contravenes a basic tenet of many on the right: that Israel must not get any smaller because the land belongs to it and because strategically that would be risky.

The result of such mix-and-match ideas is that Mr. Lieberman has drawn followers not only from the large number of Russian speakers but also from the many who are attracted to his anti-establishment tendencies, as well as the young.

“I was a supporter of Labor all my life,” said Idan Tzadok, a 24-year-old student from Haifa. “I was raised in a kibbutz, but when I came to university I woke up. I saw these people in Israel who go with the Palestinian flag.

“Peace in Israel will come when all Israelis will go to the army,” he said, adding of several Arab members of Parliament, “I am not going to pay the paycheck of these people who go and support Hamas.”

In an unscientific survey of 10 high schools across Israel, Mr. Lieberman’s party took first place followed closely by Likud, Kadima and then Labor.

Alex Miller, a member of Mr. Lieberman’s party, told the newspaper Haaretz that it was natural that the party’s message would appeal to the young.

“Loyalty is the most burning issue for the youth,” he said. “They’re about to go in the army and therefore national honor is important to them. They want someone whose word is good, who stands behind his principles. Avigdor Lieberman projects strength.”

Mr. Lieberman began his political career working for Likud, becoming campaign chairman in the 1990s for Mr. Netanyahu and director general of his office when he was prime minister. He later formed Yisrael Beitenu, winning election to Parliament as its party leader. He has held several ministerial portfolios but only for short periods because of his tendency to fall out of favor with those in power.

Lately he has also come under investigation for the business practices of a company owned by his daughter, including allegations of money laundering, fraud and breach of trust. The police have said they believe that the daughter, Michal Lieberman, was serving as a front for her father.

Mr. Lieberman said he welcomed the investigation because the more he was seen as pursued by the establishment, the more popular he became. The investigation, he said, would add four seats to his party’s take.