Prime Minister Benjamin Netanyahu failed to secure a clear election victory last month. It has put the country's longest-serving leader in an even weaker position at a time when he is preparing to face a looming indictment on corruption allegations.
Netanyahu, colloquially known as Bibi, denies wrongdoing.
The incumbent's right-wing Likud Party came second with 32 seats in the 120-member seat Knesset on September 17, while former military chief Benny Gantz's centrist Blue and White party received 33 seats.
With neither party able to secure a clear lead, the result has pushed back a long-awaited U.S.-led Israeli-Palestinian peace plan.
"The impact on the Trump peace plan is unclear," Jake Walles, who served as U.S. consul general in Jerusalem from 2005 to 2009 and is now a fellow at the Carnegie Endowment for International Peace, told CNBC via telephone.
"Certainly, the administration can't move forward until there is a new Israeli government in place. Then it will have to brief the new PM (presumably Gantz) about what's in the plan and get his approval to publish it," Walles said (parentheses in the quote are his).
"At that point, it will be close to the end of the year; will Trump still want to proceed? Who knows, but, in any case, (Palestine) will still reject it so it won't lead to a negotiating process or an agreement of any kind."
Netanyahu, who has been a close ally of Trump, had initially been expected to reveal the political portion of the so-called peace plan shortly after the mid-September election. But, his failure to secure a parliamentary majority has delayed this announcement.
Palestine had already rejected the economic component of the peace plan when it was unveiled at a conference in Bahrain earlier this year.
The U.S. plan is reportedly expected to address the question of a Palestinian state and the future status of the occupied West Bank, an issue that has remained unresolved since the 1967 war.
Right-wing parties in Israel view the West Bank as an integral part of Israel. For Palestinians, the territory is viewed as the heartland of their future state.
Past U.S. administrations have viewed Israeli settlements in the West Bank as an impediment to a possible peace deal between Israel and the Palestinians, and much of the world considers the settlements to be illegal.
President Reuven Rivlin had asked Netanyahu to try to form a unity government last week, hoping a power-sharing deal with Gantz might be possible for the two rivals to secure a parliamentary majority.
However, negotiations broke down on Sunday and, with no apparent route out of the crisis, it has raised the possibility of a third national poll over the coming weeks.
"This second election has been devastating in terms of stability and trust... But, a third election would be detrimental to the whole political system," Gayil Talshir, a political science professor at Jerusalem's Hebrew University, told CNBC via telephone.
Talshir said the main obstacle to forming a unity government was the possible indictment of Netanyahu. She said the long-time prime minister was thought to be considering taking one of three options: return to power and claim immunity to the corruption charges, ask for the corruption charges to be reduced to something less severe or strike a deal with the president to receive amnesty for all alleged political offenses.
Israel's attorney general is expected to decide whether to formally charge Netanyahu in three corruption investigations by the end of this year after a pre-trial hearing this month.
A majority in the Knesset could decide to grant Netanyahu immunity from prosecution until the end of his term as prime minister. Some of the right-wing parties in the country have signaled support for such a move, but it would probably draw public outcry and legal challenges at the Supreme Court.
"It seems to me that Netanyahu doesn't yet fully realize the situation," Talshir said.
- CNBC's Natasha Turak contributed to this report.