JERUSALEM — With the negotiations over a unity government in Israel at a critical juncture, Prime Minister Benjamin Netanyahu and his chief rival, Benny Gantz, asked the country’s president early on Tuesday to extend the deadline to form a joint coalition to midnight Wednesday.

The two sides, hampered by discord and acrimony in their discussions over a power-sharing deal, had been unable to reach an agreement before a deadline passed at midnight Monday.

The president, Reuven Rivlin, acceded to the extension request on the understanding that the two sides were close to an agreement, according to a statement early Tuesday morning from his office.

If the two sides cannot resolve their differences by Wednesday night, Mr. Rivlin is expected to refer the task of forming a government to Parliament, a step that would start a countdown toward a possible new election, Israel’s fourth in little over a year.

The Parliament would then have 21 days to come up with a candidate for prime minister endorsed by 61 members of the 120-member assembly. Failing that, Parliament would automatically disperse, sending Israelis back to the ballot box.

In separate televised statements on Monday night, just hours before the original deadline, both Mr. Netanyahu and Mr. Gantz expressed their desire for a unity government, citing the emergency created by the coronavirus pandemic, but neither explained what was standing in the way of an agreement.

“We will keep making every effort to achieve the unity that the country so needs these days,” Mr. Netanyahu said in a speech that began with a lengthy explanation of the latest coronavirus restrictions. Soon after, he called on Mr. Gantz to negotiate.

“Benny, I’m waiting for you in the prime minister’s residence in Jerusalem,” Mr. Netanyahu wrote on Twitter. “Let’s meet and sign even tonight on forming a national emergency government that will save lives and work for the citizens of Israel.”

In an impassioned speech half an hour later, Mr. Gantz, who was unable to form his own coalition despite having received more endorsements from lawmakers than Mr. Netanyahu did after the March election, exhorted the prime minister to come to an agreement.

“Netanyahu, we have arrived at the moment of truth,” he said. “The citizens of Israel expect us — both of us — to make difficult decisions.”

Last month’s election, the third inconclusive ballot since April 2019, left Mr. Gantz’s centrist Blue and White party and its allies with a slight numeric advantage in Parliament but, like Mr. Netanyahu’s conservative Likud party, it was unable to muster a majority coalition.

The only apparent way out of the deadlock was to join forces.

Citing the good of the country, Mr. Gantz reneged last month on repeated campaign promises not to sit in a government under Mr. Netanyahu, who faces trial on corruption charges. Mr. Gantz agreed in principle to enter a unity government in which Mr. Netanyahu would remain prime minister for the first 18 months and Mr. Gantz would take over for the following 18 months.

Mr. Gantz’s party immediately broke up as a result of his political reversal, diminishing his political leverage as many former supporters accused him of betraying their trust.

In his speech on Monday, Mr. Gantz defended his about-face, saying he had put the country’s interests ahead of his own and adding, “I’m at peace with myself and I’m at peace with my decision.”

“The state of emergency that has been forced upon Israel and the entire world does not allow leaders to close their eyes and ears and remain concerned only with their own egos,” he said.

But the talks, for what was meant to be a national emergency government to combat the coronavirus crisis, became bogged down in political and legal details.

Many analysts were left questioning if Mr. Netanyahu, Israel’s longest serving prime minister and a renowned political survivor, really wanted an agreement or if his plan was just to neutralize Mr. Gantz and aim for a fourth election all along.

Even if the task of forming a government passes to Parliament, Mr. Netanyahu and Mr. Gantz would still have three weeks to negotiate a power-sharing deal.

Alternatively, with the opposition in pieces, Mr. Netanyahu, who already has the endorsement of 59 lawmakers, could try to persuade a couple of defectors to cross the lines and form a narrow government.

If neither of these things happen, a fourth election would become inevitable.

So far, the unity talks have been rancorous, with Mr. Netanyahu seemingly focused on ensuring he can remain in office despite his legal troubles. Unity talks broke down last week as Mr. Netanyahu tried to change the procedures for appointing Supreme Court judges, according to Blue and White officials.

After talks resumed again a few days later, Mr. Netanyahu, analysts said, was mostly concerned with finding a way to override any future Supreme Court decision barring a candidate charged with crimes — like himself — from forming a government.

Mr. Gantz, for his part, has been trying to salvage his reputation by refusing to give way on Likud demands that he considers harmful to democracy and the rule of law.

But even if a unity government is formed, political analysts were skeptical that Mr. Gantz would get a turn as prime minister after Mr. Netanyahu, or that the government would even last that long.

“This isn’t an emergency government, nor does it have anything to do with the coronavirus,” Sima Kadmon, a political columnist, wrote in the newspaper Yediot Ahronot on Monday. “Netanyahu managed to do to Gantz what he does to everyone: first skin him, then barbecue him.”