According to an email from Rivlin's office, the president intends to hand the task of putting together the next coalition over to former military chief Benny Gantz, who leads the centrist Blue and White bloc and faces an equally hard path.
If Gantz can't form a government, which would require support from a majority of the 120-member parliament, then a majority of parliament members could nominate someone from their ranks - a move without precedent.
Gantz's Blue and White party was the largest to emerge from the 17 September ballot, though he also has no clear path to a parliamentary majority.
"In the past weeks I made every effort to bring Benny Gantz to the negotiating table, every effort to establish a broad national government, every effort to avoid another election", said Netanyahu, who turned 70 on Monday. But President Reuven Rivlin gave Netanyahu the first opportunity to form a government because he had more support, 55 seats, than any other candidate.
Netanyahu released a video Monday criticizing Gantz's supposed unwillingness to allow Likud to take the lead.
In contrast to Mr Netanyahu, whose political career spans three decades, the 60-year-old Mr Gantz is a newcomer who only burst onto the scene over the last year.
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By "liberal", it signalled it would seek to limit the influence of religious parties in forming a coalition.
"The time of spin is over; it is now time for action", his party said in a statement posted to Twitter.
Without Likud, Mr Gantz will have a hard time securing a majority in parliament.
Now Gantz, too, will have 28 days to make good on his mandate.
If Israelis go to the polls again it would be the third time in a year, after two inconclusive election results pushed the country into a political crisis.