The two largest parties-Netanyahu's reigning right-wing Likud Party, and left-wing challenger Blue and White, led by Benny Gantz and Yair Lapid-are almost tied, though some 10 percent of the votes have yet to be counted.
For the Palestinians though, there may be little difference between hawkish Netanyahu and centrist former armed-forces chief Gantz. "That's why we're the biggest party in Israel".
Yemen's Houthi group said on Wednesday it had identified dozens of sites in the United Arab Emirates as possible targets, . "But the responsibility for this, as well as the responsibility for forming a government that serves all the citizens of Israel with the dedication it deserves, is yours - the elected officials, and especially the heads of the two major parties".
Joint List Knesset member Ahmed Tibi said on Wednesday that "we won't and can not take any part in the government or the coalition", but added that "there are other ways to advance what we want as a joint ticket, to influence the decision-making process, and to be a significant actor in the parliament and in politics".
Rivlin will be holding meetings with party heads in the coming days to see whom they recommend to head the government.
Netanyahu signed an agreement with three right-wing and religious parties to negotiate a coalition deal as one bloc of 55 MKs, but then called on Blue and White to begin immediate negotiations without preconditions.
Despite Gantz's slim lead, neither has an obvious path to form a majority coalition.
Both parties would require partners to establish a government with a ruling majority.
But Tuesday's vote didn't go well for Benjamin "Bibi" Netanyahu, the longest-serving prime minister in Israeli history, with 12 years in office.
"Almost certainly this, what we can call a broad mainstream unity government will exclude the hard ideological camps on the right and the left", said analyst Yossi Klein Halevi of the Hartman Institute.
Politicians will have 28 days to try and form a government.
Gantz added: "We won't give in to an imposed [coalition]".
Israel is the only true democracy in the Middle East. Here are the latest voting results from the Israel election.
Lieberman has successfully bolstered his popularity in the September elections, securing nine mandates.
At the same time as calling for Gantz to join him in forming that government, Netanyahu worked to shore up support on his right.
"We can not and there is no reason to go to third elections", he added. Netanyahu proposed a rotating premiership among the two leaders. Likud and its natural allies of religious and ultra-nationalist parties have fallen several seats short of the needed majority. The fiercely loyal Likud is unlikely to oust Netanyahu.
The Jerusalem Post reports that during a memorial service for Simon Peres, Netanyahu said: "Let's cooperate like Peres and [former prime minister Yitzhak] Shamir".
Complicating matters further is a rotation arrangement already in place within the Blue and White Party, whereby party co-leader Yair Lapid would become prime minister should Blue and White lead the country for two-and-a-half years.
The next step in the battle to lead Israel will be for President Reuven Rivlin to designate one candidate to try to assemble a coalition government.