In a first, the presidential statement said Rivlin has ordered the consultations be open to public scrutiny.
"The meetings with the parties will be broadcast live on all platforms, to ensure transparency for Israeli citizens," it said in English.
The talks at the president's official Jerusalem residence will kick off with Netanyahu's Likud party, followed by the Blue and White alliance of his defeated centrist challenger, former military man Benny Gantz.
Likud came out just ahead of Blue and White in the general elections with 36 seats in the 120-member parliament compared to 35 for its rivals.
Meetings with smaller parties will continue throughout Monday and Tuesday, the statement said, adding that "at the end of the round of consultations" Rivlin will announce who he will invite to try to forge a viable coalition.
The president must make his decision by April 24 at the latest.
The chosen politician then has 28 days to form a government, but can ask for an extension of up to 14 days.
An initial count of the votes cast put Likud and Blue and White neck and neck with a projected 35 seats each.
However, an update issued just before midnight Thursday showed Likud a whisker ahead, with local media saying that translated into a one-seat lead, making it the largest party in parliament.
The Central Elections Committee, which updated the figures to include voting by soldiers on active service, prisoners, diplomats abroad and seamen, warned the results could still change until the final and binding tally is published on April 17.
The combined total of seats projected for Likud and what Netanyahu calls its "natural partners" among the right and religious Jewish parties adds up a comfortable 65 seats -- a workable 5-seat majority.
No single political party in Israel's more than 70-year history has ever won an absolute majority in elections and coalitions are the norm.
Gantz and his potential allies trail far behind and on Friday he called Netanyahu to congratulate him and send greetings for next week's Passover Jewish festival.
"I would like to congratulate you on your achievement in the elections," a Blue and White statement quoted Gantz as saying.
Maariv daily, citing an unnamed senior official, said Rivlin could suggest Netanyahu and Gantz agree to form a national unity government.
Such an alliance would have a strong majority and would not be beholden to smaller parties.
Contentious issues on the horizon include a pending initiative from US President Donald Trump to try to resolve the conflict with the Palestinians, public spending cuts or a tax hike and the fierce debate over whether ultra-Orthodox Jews should perform mandatory military service.
Commentators have said the looming prospect of indictments in several corruption cases against Netanyahu makes him vulnerable to demands for concessions from smaller parties in any coalition bargaining.
Israel's attorney general has announced his intention to charge him with bribery, fraud and breach of trust in three cases.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)