Israeli Prime Minister Benjamin Netanyahu and his rival Benny Gantz are jostling over the terms of a unity government, after the country's second election in a year ended in deadlock.
Local media say both men's parties failed to win enough votes on Tuesday to build a coalition with a majority.
Mr Netanyahu urged Mr Gantz to start immediate negotiations to form a unity government.
Mr Gantz said he wanted a unity government - but only one led by him.
His party has ruled out joining a coalition led by Mr Netanyahu, who faces possible corruption charges.
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At a ceremony in Jerusalem on Thursday, President Reuven Rivlin welcomed the prime minister's "important call".
Mr Rivlin will hold consultations with party representatives before nominating a candidate whom he believes has the best chance of forming a government.
What was the outcome of the election?
Official results have been slow to be released by the Central Election Committee, with only 68.6% of votes counted by 14:56 (11:56 GMT) on Thursday.
Mr Gantz's Blue and White alliance is 0.77% ahead of Mr Netanyahu's right-wing Likud party. The Joint List, an alliance of Arab parties, is in third place, the ultra-Orthodox Shas party in fourth, and the nationalist Yisrael Beiteinu party in fifth.
The CEC tally does not say how this translates into seats in the 120-seat Knesset (parliament) but Israeli media reported that Blue and White was on track to win 33 seats, two more than Likud.
A centre-left bloc led by Mr Gantz was projected to control 57 seats and a bloc of right-wing and religious parties allied to Mr Netanyahu looked set to have 55.
Neither would therefore be able to form a majority coalition without support from Yisrael Beiteinu, whose leader Avigdor Lieberman has called for a "broad, liberal" unity government.
Mr Lieberman prevented Mr Netanyahu from forming a majority coalition after the last election in April because he refused to back down in a longstanding dispute with religious parties over exempting ultra-Orthodox men from military service.
How have the party leaders responded?
On Wednesday night, officials said the prime minister had cancelled a planned trip to New York next week to address the United Nations General Assembly.
In a video released on Thursday, Mr Netanyahu said that, "to his regret", the election results showed he would not be able to establish a right-wing government.
"There is no choice but to establish a broad government, as broad as possible, composed of all those elements to whom the state of Israel is dear," he added.
"Benny, we must set up a unity government today. The people expect the two of us to show responsibility and work for co-operation. That is why I call on you, Benny. Let's meet today, at any hour, at any time, to put this in motion."
The BBC's Tom Bateman in Jerusalem says the move is characteristic of Mr Netanyahu - attempting to grab the initiative amid the political deadlock, while Israel's president has to task one party leader with trying to form a coalition.
Mr Netanyahu also referenced the unity government between Shimon Peres and Yitzhak Shamir in the 1980s - which some local media saw as a hint towards a possible rotating premiership.
Mr Gantz gave a statement later on Thursday, saying he intended to form a unity government "for all people of Israel", led by himself.
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"Blue and White has at the time I am speaking won 33 seats, while Netanyahu has not obtained a sufficient majority to form a coalition as he hoped," he said.
"The government which will be established has to be a just one," he added, insisting that unity governments had to be formed "not from political blocs and spin, but from responsibility and seriousness".
Mr Netanyahu has already allied his Likud party with religious and right-wing coalition partners which are unlikely coalition bedfellows for Mr Gantz's Blue and White party.
One source in Blue and White told Israeli media that they saw Mr Netanyahu's offer as a PR stunt.
The source added that the party would not join forces with Likud while Mr Netanyahu remained leader because of the charges of bribery, fraud and breach of trust he faces in connection with three corruption cases, pending a final hearing next month. He denies any wrongdoing. Source: bbc