On Monday, President Rivlin summoned Prime Minister Netanyahu and his election rival Gantz to his official residence, hoping to break a political deadlock that threatens to force a third election in less than a year.
Representatives of Israel's two largest parties are meeting on Tuesday to discuss the possibility of forming a unity government after last week's indecisive elections.
The meeting comes a day after Prime Minister Benjamin Netanyahu of the Likud party and his rival Benny Gantz of the Blue and White party held their first talks after the September 17 polls.
Gantz had been reluctant to initiate talks with Netanyahu, but Reuven Rivlin, Israel's president, intervened on Monday and summoned the two politicians to his official residence. President Rivlin expects the two parties to break the political impasse, otherwise the country could be forced to hold a third election in less than a year.
The president is constitutionally responsible for nominating a prime minister after national elections. But neither Likud nor the centrist Blue and White party secured enough seats with their allies to form a majority.
Last week's vote marked the first time Israel has held two votes in one year. The ballot was called after Netanyahu's right-wing Likud party failed to win enough votes to form a coalition following elections in April.
Israeli media have reported that the Blue and White party won 33 out of 120 parliamentary seats; Likud has 31.
Boost for Gantz
Last week, Netanyahu offered to form a unity government, but it was rejected by former army general Gantz. Instead, Gantz expressed his wish to become prime minister.
"If Netanyahu moves aside, we'll have a unity government," announced Gantz's second-in-command, Yair Lapid.
Meanwhile, Israel's Arab bloc, the Joint List party, has announced its endorsement of Gantz's party.
By supporting the Blue and White leader, the Arab bloc broke with its own precedent of withholding support for either of the top Israeli candidates.
"We want to bring an end to the era of Netanyahu, so we recommend that Benny Gantz be the one to form the next government," party head Ayman Odeh said Sunday.
Odeh's party saw its turnout swell in last week's election. With 13 seats, the Joint List party is now the third-largest grouping in the Knesset.
Netanyahu blasted the Arab party's support for Gantz, saying that Israelis now were faced with two choices: "A minority government that leans on those that reject Israel as a Jewish, democratic state," or a "broad national government" composed of his and Gantz's parties.