According to a timetable released by the President's Residence, the consultations, which are being broadcast live, are expected to be concluded by Sunday evening.
Rivlin will hear the recommendations of eight factions that passed the electoral threshold before announcing whom he will appoint with forming the new government.
The consultations are an official procedure in Israel after elections, in which the 120 newly-elected members of the parliament give their recommendations on which leader should form the government.
At no time in the Israeli history did any party win an outright majority to form a government, as a coalition needs at least 61 seats out of the 120-member parliament.
The election results on March 2 led to another de facto political deadlock like two previous inconclusive rounds of votes in 2019.
Israeli Prime Minister Benjamin Netanyahu is supported by his own right-wing Likud party and a bloc of a pro-settler party and two Jewish ultra-Orthodox parties, with a total seats of 58.
His main rival, Benny Gantz, Israel's former military chief and leader of the centrist Blue and White party, has been generally supported by left-wing and center-right parties but it is still not clear how many of them will actually give their support to him.
On Sunday morning, Netanyahu repeated his call for "an emergency" unity government with the Blue and White to fight the COVID-19 crisis.
"Faced with the global and national emergencies, we must unite forces and form a strong and stable government that can pass a budget and make tough decisions," he wrote on Twitter.
Netanyahu, who is facing criminal charges in three separate corruption cases, said he will act as a prime minister of the proposed government for two years before he would be replaced by Gantz.
Gantz, however, dismissed the offer as a spin. "Netanyahu, let's not manipulate the public," he wrote on Twitter.
It is worth noting that Netanyahu's trial, which was supposed to begin on Tuesday, was postponed to May 24 in the wake of a "state of emergency" in the entire Israeli court system announced by Netanyahu's justice minister after midnight on Sunday.