JERUSALEM — Prime Minister Benjamin Netanyahu of Israel said on Monday that he would be willing to hand over leadership for one year to a longtime right-wing rival, Naftali Bennett, in a last-ditch effort to cobble together a new government.

Mr. Netanyahu, who has spent the last 12 years in office and is now standing trial on corruption charges, announced the offer just ahead of a deadline to form a government, in the wake of Israel’s fourth inconclusive election in two years.