In August of that year, Mr. Duterte on live television accused a series of politicians and other public figures of being involved in drugs, urging them to turn themselves in. Chief Justice Sereno, whose duties include overseeing more than 2,000 lower courts, said that judges on the list should not surrender to the authorities unless they were shown a warrant duly issued by a local court.

Mr. Duterte responded by saying the chief justice “must be joking.” He added: “If this continues, you’re trying to stop me, I might lose my cool,” he said.

Mr. Duterte’s drug war, which has left thousands of Filipinos dead and been widely condemned by rights groups, has yet to come before the Supreme Court. But Chief Justice Sereno has voted against at least two of the president’s other high-profile initiatives: his declaration of martial law in the southern Philippines last year, after Islamist militants took over the city of Marawi, and his move to give the dictator Ferdinand Marcos a hero’s burial decades after his death.

She said that Mr. Duterte, an admirer of Mr. Marcos, had “acted with grave abuse of discretion” when he ordered the dictator’s remains transferred to the Heroes Cemetery in Manila in 2016. She said the president was shirking “obligations to do justice for human rights victims” under the Marcos regime, which a popular uprising overthrew in 1986.

The court sided with Mr. Duterte in that case, and on the martial law issue. But Chief Justice Sereno’s supporters say she is an impediment to authoritarian policies, with one calling the impeachment drive a “well-orchestrated but pathetic telenovela.”

The chief justice said that nothing had emerged from the hearings to justify her removal.

But Mr. Duterte’s spokesman, Harry Roque, has called on her to resign “to protect the institution.”