TEL AVIV — “He’s a magician! He’s a magician!”

It was nearly 2 a.m. on April 10, and supporters of Prime Minister Benjamin Netanyahu, who had packed a sporting arena in Tel Aviv, were cheering his apparent election victory.

Facing a stiff challenge from a former military chief and the prospect of criminal indictment on corruption charges, Mr. Netanyahu, whose unrivaled political instincts had him on track to become Israel’s longest-serving prime minister, had pulled another rabbit out of his hat.

Less than two months later, after his failure to assemble a governing coalition forced the country into an unprecedented do-over election, Israeli pundits are asking whether even magic can save him now.

With the new election still more than three months away, no one is foolish enough to write him off altogether. But the math and the calendar are unforgiving.