He will have 28 days to form a coalition and can ask Rivlin for a two-week extension if necessary.
Israel's president tasked Prime Minister Benjamin Netanyahu on Wednesday with assembling a new government after power-sharing talks with his strongest rival, Benny Gantz, failed following an inconclusive election.
Netanyahu, head of the right-wing Likud party, and Israel's longest-serving leader, still has no clear path to a fifth term after emerging from the Sept. 17 ballot, the second this year, short of a parliamentary majority.
"I have decided to give you, sir, the opportunity to assemble a government," President Reuven Rivlin said to Netanyahu at a nomination ceremony.
He will have 28 days to form a coalition and can ask Rivlin for a two-week extension if necessary. Netanyahu's failure to clinch victory in a ballot in April led to last week's election and left him politically weakened.
In the new countdown, Likud has the pledged support of 55 legislators in the 120-member parliament, against 54 for Gantz's centrist Blue and White Party. The two parties failed to reach a coalition deal in talks launched on Tuesday.
Former Defence Minister Avigdor Lieberman, a possible kingmaker, has been keeping his far-right Yisrael Beitenu party on the fence since the Sept. 17 ballot, citing differences with both Likud's and Blue and White's political allies.