Gantz had not yet responded, but he has repeatedly called for a unity government.
His Likud along with its right-wing and religious allies won a majority, but he failed to form a coalition and opted for a second election rather than risk having Rivlin choose someone else to try.
"The president will be guided by the need to form a government in Israel as quickly as possible and to implement the will of the people as expressed in the results of the election, as well as the need to avoid a third general election", Rivlin tweeted on Wednesday. "Therefore, there is no choice but to establish a wide unity government as wide as possible that's made up of all the officials that Israel called on", The Jerusalem Post quoted Netanyahu as saying. And I call on you, Benny.
"The nation expects us, both of us, to show responsibility and act in cooperation", he said. Let's meet today. At any time, at any moment.
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PA Minister of Civil Affairs Hussein al-Sheikh tweeted that: "After the initial results of the Israeli elections had been announced, we reaffirm that the Israeli elections and its results are an internal Israeli affair".
Both a Gantz-led Centre-Left and a Netanyahu-led Right-wing coalition do not have enough men to conquer the helm.
With Israeli media reporting more than 90 percent of votes counted in Tuesday's election, the bloc led by Netanyahu's Likud party was more or less even with a likely grouping headed by Gantz's centrist Blue and White party.
"When there was no clear victory in Knesset (parliamentary) elections, Shimon opted for national unity. He and Yitzhak Shamir agreed to cooperate - to navigate Israel's path to safety", the Likud leader said.
Iran's Khamenei rejects talks with United States
President Donald Trump warned the USA was " locked and loaded " to respond to the attacks, as Breitbart News reported . That moment aside, the Trump administration has seemed willing-and eager, at times-to start a war with Iran.
"Let's cooperate like Peres and [former prime minister Yitzhak] Shamir", Netanyahu said. The prime minister has been seeking to win majority support for legislation granting him immunity.
There was no immediate response from Blue and White.
If Netanyahu is indicted, by law, he must resign.
Shas leader Aryeh Deri gave his consent via phone and said he would sign the document soon. He failed when Liberman refused to join unless a bill formalizing exemptions to mandatory military service for yeshiva students was passed as is, a demand flatly rejected by the premier's ultra-Orthodox coalition partners.
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With no clear victor, some analysts here see the chances of a coalition government coming in place quite bleak at the moment. In approximately two weeks, Netanyahu will find himself facing a hearing by Israel's Attorney General Avichai Mandelblit on October 2, and he will not have in coalition in place that can provide him with legal protection against indictment. With neither bigger party set to get a clear majority, even with their coalition, Israel faces the prospect of a long period of political haggling once again. Contesting parties have found themselves in a Catch 22 situation. He said, "In the next few days, we will begin negotiations to form a strong, Zionist government and in order to avert a unsafe, anti-Zionist government", a reference to the Blue and White, which he denounces for being ready to work with the MPs from the Arab Joint List.