The last time an Israeli politician beside him had the chance to form a government was in 2009, when Tzipi Livni, then the foreign minister, narrowly edged Mr. Netanyahu in an election. But she failed to muster a majority and Mr. Netanyahu succeeded, completing a comeback after having served a previous term as prime minister in the late 1990s.

Mr. Gantz, a career soldier making his first run for office, tied with Mr. Netanyahu in their first contest in April, but Mr. Netanyahu had more supporters in Parliament and was given the chance to form a government. He appeared well on his way to a fourth consecutive term only to be thwarted by a surprise defection by Mr. Liberman.

Rather than let Mr. Gantz be given a chance, Mr. Netanyahu orchestrated a second election, held on Sept. 17.

Mr. Gantz narrowly edged Mr. Netanyahu in that election, but Mr. Netanyahu’s coalition of right-wing and ultrareligious parties again came away with a larger bloc in Parliament than Mr. Gantz’s alliance of center-left parties. Once more, Mr. Netanyahu was handed the first attempt at forming a government.

Mr. Netanyahu may still have another path back to the premiership: If Mr. Gantz cannot form a government within his allotted time, the president can hand the task to Parliament, giving lawmakers an additional 21 days to come up with a candidate who can command a majority. Mr. Netanyahu may be hoping, at that point, that the public and political pressure to avoid a third election will persuade the half-dozen additional lawmakers whose support he needs to come to his side.

Analysts have also speculated that Mr. Netanyahu may prefer a third election, perhaps believing that the attorney general would ultimately drop the bribery indictment, the heaviest of three charges he is facing. Under such an outcome, Mr. Netanyahu could claim a degree of vindication and campaign while facing lesser charges of fraud and breach of trust, and insisting that they, too, would come to naught in court.

The gamesmanship between Mr. Netanyahu and Mr. Gantz since the election last month has resembled a chess match in which Mr. Netanyahu’s position was weaker than after the April election but the conclusion was hardly foregone.