Senator Lindsey Graham, the South Carolina Republican who died this month at 71, worked privately with Prime Minister Benjamin Netanyahu of Israel to coordinate a bipartisan group of lawmakers to denounce the International Criminal Court investigation into Mr. Netanyahu just before the 2024 election, footage of their discussions shows.

The conversations were among roughly 400 hours recorded by the British filmmaker Alex Holder, who trailed Mr. Graham for roughly four years for a planned documentary project titled “Lindsey’s Game.” The footage of two discussions with Mr. Netanyahu about rebuking the I.C.C. was reviewed by The New York Times.

The calls took place just days before the 2024 election, on Oct. 29, as Mr. Graham was in Ohio, where he was campaigning with Donald Trump Jr. for Bernie Moreno in a Senate race, according to Mr. Holder.

The discussions underscore the significance of the relationship between Mr. Graham and Mr. Netanyahu and its impact, including with the senator’s colleagues. The two men were days away from an election that would return Donald J. Trump to the White House. Mr. Graham was a valuable interlocutor for Mr. Netanyahu even before that, but the call took place as the prime minister was prepared to discuss with Mr. Trump waging an increasingly expansive war against Iran.

Mr. Netanyahu was in Washington on Tuesday for Mr. Graham’s funeral, as well as a meeting with Mr. Trump about the war in Iran, which is becoming an intractable problem for the president.

Mr. Netanyahu remains the subject of an arrest warrant by the I.C.C., but Israeli officials have called the effort politically motivated. Mr. Netanyahu has made foreign trips since it was issued, and is set to travel to the U.N. General Assembly in New York in September.

Mr. Trump has declared that he will not be arrested, despite Mayor Zohran Mamdani calling Mr. Netanyahu a war criminal and saying he had reviewed whether city officials had the power to make an arrest. The mayor later said officials concluded they did not.

In the first call with Mr. Netanyahu, which was recorded on video, Mr. Graham is seen and heard talking to the prime minister, whose voice is occasionally garbled as it comes through the line and is picked up on the senator’s microphone. But Mr. Netanyahu could be heard clearly as he discussed a pending decision by the I.C.C. about whether to seek his arrest for alleged war crimes, as a prosecutor with the court had recommended earlier in 2024.

“A three-judge panel,” Mr. Netanyahu said in the first call with Mr. Graham, will be “making the decision very soon.” He noted that the court’s chief prosecutor, Karim A. A. Khan, had faced allegations of sexual misconduct, and suggested to Mr. Graham that “the great cloud that it shed” could help delay the arrest warrant (Mr. Khan was removed from his position last week, the court’s oversight body announced.)

“It is possible to have to appeal to the court to say, ‘OK, we want to review this before you do anything, before you issue an arrest warrant. There has to be a review about this entire process,’” Mr. Netanyahu said, adding that “this cloud or this stain has to be removed” before moving forward.

“And I suggest that you send a letter, with, maybe some other senators,”Mr. Netanyahu said. When Mr. Graham offered up Speaker Mike Johnson as the “perfect person” to sign such a letter, Mr. Netanyahu replied, “Would he get Democrats?”

Mr. Graham said that he would try for a bipartisan approach.

Mr. Holder recorded the audio, without video, of a second call between the two men that he said took place later that day. Only Mr. Graham’s voice could be heard.

On that call, Mr. Graham joked to Mr. Netanyahu that he should “pay me for all this. I should get tons of shekels.” It would take “more than dinner,” he added.

He then said that “we got them both,” appearing to refer to Senators Richard Blumenthal of Connecticut and Ben Cardin of Maryland, two Democrats, who said they would sign a letter protesting the efforts to move ahead with an arrest warrant.

“So here’s why I’m calling you,” Mr. Graham told Mr. Netanyahu, saying he hoped to get the letter “out the door Wednesday,” appearing to refer to the next day, October 30. “So send me over whatever you want to say. And they want to say two things” to the I.C.C.

One was that the prosecutors should speak with the Israeli government and not move ahead with the case until “the cloud is lifted” from the prosecutor’s own problems.

What Mr. Graham requested in exchange, he said, was for Mr. Netanyahu “to call Blumenthal and Cardin and thank them for being fair-minded on this, and also thank them for trying to help with Saudi-Israel normalization. You appreciate it. You’re very interested in it, and that will encourage both of them.”

The letter was released on Nov. 1, 2024, and included Mr. Blumenthal and Mr. Cardin as signers. In addition to noting the allegations against Mr. Khan, the letter said that the I.C.C. did “not comply” with the law by declining to keep a planned meeting with Israeli representatives before applying for the arrest warrant.

A spokesman for Mr. Netanyahu did not respond to an email seeking comment on the calls.

Mr. Blumenthal declined to address questions about the calls Mr. Graham had with Mr. Netanyahu. An aide to Mr. Cardin did not respond to an email seeking comment.