An inconclusive ballot on Sept. 17, the second this year, has created a political deadlock and left Israel's longest-serving prime minister weakened.
Netanyahu, facing a looming indictment on corruption allegations he denies, has failed to secure a clear election victory twice in six months. His right-wing Likud party came second with 32 seats in the 120-member parliament, against 33 for former military chief Benny Gantz's centrist Blue and White party.
With neither party leader appearing able on his own to put together a coalition with a ruling majority, President Reuven Rivlin on Wednesday tasked Netanyahu with forming the next government in the hope of securing a power-sharing deal between the Blue and White party and Likud.
If Netanyahu fails to clinch a deal in the 28 days allotted to him, Rivlin is then likely to ask Gantz to try to form a government, though he, like Netanyahu, also has no clear path to power.
The parties' negotiators met on Friday without success, with each side blaming the other for the stalemate. Likud said Netanyahu then telephoned Gantz and the two men would probably meet on Wednesday after another round of talks.
"Prime Minister Netanyahu is making a last effort to bring about a breakthrough in the talks for a unity government," Likud's spokesman said.
The Blue and White party said it would spare no effort to form "a broad, liberal, unity government".
In a video posted on Twitter, Netanyahu said there was no other solution than for him to serve first as prime minister and Gantz to serve second within a unity government.
Gantz, however, has pledged not to serve in a government under a premier facing criminal charges. His party has accused Likud of stalling in the talks in the hope of triggering yet another election.
Netanyahu, for his part, has said that the Blue and Whites were trying to instigate a coup within Likud that would see the party join a unity government with the Blue and White party after casting aside its veteran leader.
Attorney General Avichai Mandelblit is due to hold a pre-trial hearing this week on his announced intention to indict Netanyahu on fraud and bribery charges in three corruption cases.
Netanyahu, who says he is a victim of a political witchhunt, can argue at the session against prosecution.