Netanyahu's right-wing Likud party won 30 seats in the March 17 general election and has so far signed coalition agreements with two ultra-Orthodox Jewish parties and the centre-right Kulanu led by Likud defector Moshe Kahlon.
That gives him the backing of 53 seats, still woefully short of a majority in the 120-seat Knesset or parliament.
If Netanyahu is unable to form a coalition by tomorrow midnight , President Reuven Rivlin must then assign another party leader to the task, with a 28-day deadline.
If that fails, he must select a third person who has just 14 days to complete the task. And if that also ends in failure, Rivlin would call a new election.
Incumbent Foreign Minister Avigdor Lieberman dropped a bombshell on Monday when he announced that his six-seat Yisrael Beitenu party would not be part of the next coalition.
Netanyahu now needs to sign up the far-right Jewish Home party of Naftali Bennett, which has eight seats, to achieve even a minimum workable majority of 61.
"The responsibility for forming a nationalist government rests now on Naftali Bennett's shoulders," Likud said in a statement on Monday night.
It said Jewish Home had been offered several cabinet portfolios including education, agriculture, the post of deputy defence minister and a seat on the powerful security cabinet.
Jewish Home argued that if it enabled formation of a 61-member coalition Netanyahu would soon seek to bring in the centre-left Zionist Union, gambling that a broader base would prove more stable, despite the displeasure of his own right flank.
"Since that's the case, we demand beyond the education and agriculture ministries another ministry with dramatic influence on national decision-making junctions, which would leave us with a hand on the wheel even when the left joins the government," Jewish Home Director General Nir Orbach wrote in a message to party activists yesterday.
"It won't collapse tomorrow or the next day, but in our current system of government, it's clear that a coalition of 61... will have trouble functioning for long," Yossi Verter wrote in Haaretz newspaper.
"Let's see it pass the far-reaching reforms promised by Kahlon. It certainly won't serve out its term - which it seemed almost certain to do before Lieberman, with a big grin, dropped his bombshell yesterday (Monday)."
Today, Jewish Home refused to comment on the state of negotiations or say if meetings with Likud were planned during the day.
Maariv newspaper's Ben Caspit also maintained that Kahlon, who campaigned on a platform of banking and housing reform, was likely to see his ambitions unrealised in the face of such a large parliamentary opposition.
"The really big loser (in addition to Netanyahu) from Lieberman's suicide bombing attack yesterday, is Moshe Kahlon," Caspit wrote.
"In a 61-member coalition, the chances of his passing reforms approach zero."
Before Lieberman bailed out, Netanyahu had hoped to forge a rightwing religious line-up with a majority of 67 of parliament's 120 seats.
During negotiations, Lieberman had reportedly laid down a number of far-reaching demands for his agreement to join the coalition, including full responsibility for dialogue with Washington.
He also demanded that the government adopt as a strategic goal the removal of the Islamist Hamas movement as the de facto power in Gaza.
Speaking yesterday, Lieberman said it was clear that Israel's next government had "no intention of overthrowing the Hamas regime".
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