Israel’s enemies have begun seizing on Mr. Netanyahu’s legal predicament: In Munich on Sunday, the Iranian foreign minister, Mohammad Javad Zarif, pointedly alluded to Israel’s “domestic corruption” problem in accusing Mr. Netanyahu of “aggression” to distract attention from his political troubles.

Back at home, opponents from the Israeli left and center are demanding that Mr. Netanyahu resign or declare himself “incapacitated”: Yair Lapid of the centrist Yesh Atid party, calling in vain for a no-confidence vote, said Monday that Mr. Netanyahu should appoint a temporary prime minister from within his own party.

“Israel deserves a full-time prime minister who is not engaged in anything else,” Mr. Lapid said.

Another of Mr. Netanyahu’s main challengers, Avi Gabbay of Labor, said that the prime minister had “become a liability for the citizens of Israel.” Mr. Gabbay lamented Mr. Netanyahu’s “sickly obsession for ‘what will people say’ and what will be written about him in the media,” adding, “Every day that he stays in office is damage to the country.”

Yet, Mr. Netanyahu’s right-wing coalition appears solidly behind him. On Tuesday morning, even as reports were first surfacing about the allegations of an attempt to bribe a judge, the prime minister’s coalition allies were loudly denouncing the police commissioner as biased and unprofessional, while accusing him of leaking like a sieve about the various Netanyahu-related inquiries.

At the center of the two new allegations is a close adviser to the prime minister, Nir Hefetz, a veteran Israeli journalist and political operative who in recent years has bounced back and forth between editing jobs and tending to the public image of the prime minister and his family. In 2015 he was a top strategist for the Likud Party’s successful election campaign.

Late that year, according to the police and Israeli news reports, Mr. Hefetz, working as the Netanyahu family’s media adviser, passed a message through an intermediary to Israel’s commissioner for prosecutorial oversight, Judge Hila Gerstel: Would she drop a corruption case against Mr. Netanyahu’s wife, Sara, in exchange for being named attorney general?