The two-day round of consultations began with a meeting with Benny Gantz, leader of the Blue and White party. On Tuesday's elections, the centrist party won a narrow victory over Prime Minister Benjamin Netanyahu's right-wing Likud party.
The meeting will be followed with talks with leaders of the Likud, the Arab-Jewish Joint List, the Jewish ultra-Orthodox party of Shas, and the far-right party of "Israel Our Home," whose leader, Avigdor Lieberman, has called for "a broad, liberal unity government" with the parties of Blue and White and Likud.
The president is scheduled to meet with leaders of other smaller parties on Monday.
Rivlin will hear their recommendations before he appoints the one to form the new government.
The consultations are an official procedure in Israel after elections, in which the 120 newly-elected members of the parliament, or the Knesset, give their recommendations on which leader should form the government.
Last week, Rivlin vowed he will do "every effort" to avoid a third election cycle. In a public remark on Thursday, he welcomed a unity government composed of the two largest parties.
In the voting on Tuesday, Blue and White won 33 seats, becoming the largest faction in the Knesset. Netanyahu's Likud won 31 seats, casting doubts on the ruling of Israel's longest-serving prime minister.
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