For years, Prime Minister Benjamin Netanyahu of Israel has been standing trial for corruption, even as he oversees conflicts in Gaza, Lebanon and Syria.

Now, he is locked in a separate standoff with Israel’s law enforcement authorities, a police investigation that has renewed opposition calls for his resignation.

Known as “Qatargate,” the case hinges on the claim that Mr. Netanyahu’s media advisers were paid by a representative of Qatar to promote Qatari interests in the Israeli news media. That claim has angered many Israelis because it created the perception of a conflict of interest; for years, Qatar has hosted leaders of Hamas.

Mr. Netanyahu’s recent efforts to fire the head of the Shin Bet, the Israeli internal security agency, has compounded the furor. It was the Shin Bet that first instigated the investigations into Mr. Netanyahu’s aides.

Mr. Netanyahu has dismissed the case as a “political manhunt” aimed at toppling him from power. A Qatari government official said it was a “smear campaign” aimed at distracting from Qatar’s efforts to mediate between Israel and Hamas.