ATLANTA — Six days after major news organizations declared Donald J. Trump the loser of the 2020 presidential election, his allies were applying a desperate full-court press in an effort to turn his defeat around, particularly in Georgia.

The pro-Trump lawyer Sidney Powell went on television claiming that there was abundant evidence of foreign election meddling that never ultimately materialized. Another lawyer, L. Lin Wood, filed a lawsuit seeking to block the certification of Georgia’s election results.

That same day, Nov. 13, 2020, Senator Lindsey Graham, the South Carolina Republican and one of Mr. Trump’s most ardent supporters, made a phone call that left Brad Raffensperger, Georgia’s Republican secretary of state, immediately alarmed. Mr. Graham, he said, had asked if there was a legal way, using the state courts, to toss out all mail-in votes from counties with high rates of questionable signatures.

The call would eventually trigger an ethics complaint, demands for Mr. Graham’s resignation from the left and a legal drama that is only culminating now, nearly two years later, as the veteran lawmaker fights to avoid testifying before an Atlanta special grand jury that is investigating election interference by Mr. Trump and his supporters.

Mr. Graham has put together a high-powered legal team, which includes Don F. McGah II, a White House counsel under Mr. Trump. While Mr. Graham’s lawyers say that they have been told that he is only a witness — not a target of the investigation — that could change as new evidence arises in the case, which is being led by Fani T. Willis, the district attorney of Fulton County, Ga. Her efforts to compel Mr. Graham to testify have been aided by legal filings from a number of high-profile, outside attorneys, including William F. Weld, a Trump critic and former Republican governor of Massachusetts.

Underscoring the risks for Mr. Graham, lawyers for 11 people who have been designated as targets who could face charges in the case have said that they were previously told that their clients were only “witnesses, not subjects or targets,” according to court filings.

On Sunday, the U.S. Court of Appeals for the 11th Circuit temporarily blocked Mr. Graham from testifying and directed a lower court to determine whether he was entitled to a modification of the subpoena based on constitutional protections afforded to members of Congress. After that, the appeals court said, it will take up the issue “for further consideration.” The matter is now back before Leigh Martin May, a Federal District Court judge who already rejected Mr. Graham’s attempt to entirely avoid testifying; she asked the sides to wrap up their latest round of legal filings by next Wednesday. It seems increasingly likely that Mr. Graham will testify next month.

Ms. Willis has said that she is weighing a broad array of criminal charges in her investigation, including racketeering and conspiracy. She has already informed at least 18 people that they are targets, including Rudolph W. Giuliani, Mr. Trump’s former personal lawyer. Mr. Giuliani fought to avoid testifying in person but was forced to appear before the grand jury last week.

Regarding Mr. Graham, Ms. Willis’s office is seeking to learn more about his role in Mr. Trump’s post-election strategy, and who he spoke to on the Trump campaign team before or after he called Mr. Raffensperger. While Mr. Trump assailed Mr. Raffensperger on Twitter as a “so-called Republican” on the same day as that call, Mr. Graham told CNN that the former president did not encourage him to place the call.

Mr. Graham has insisted that he did nothing improper, and his lawyers have argued that a sitting senator should not have to answer questions about his conduct in a state court.

“We will go as far as we need to go and do whatever needs to be done to make sure that people like me can do their jobs without fear of some county prosecutor coming after you,” Mr. Graham said recently.

That Mr. Graham finds himself sparring with the district attorney because of his involvement with Mr. Trump might have seemed unlikely before the 2016 election, when he called Mr. Trump “unfit for office” and “a race-baiting, xenophobic religious bigot.” But he swiftly became a key ally of Mr. Trump and spent the days after the 2020 election railing against its legitimacy. “If Republicans don’t challenge and change the U.S. election system, there will never be another Republican president elected again,” he told Fox News on Nov. 9, 2020. Democrats win, he said, “because they cheat.”

At the time he made the call to Mr. Raffensperger, Mr. Graham was chairman of the Senate Judiciary Committee, and he has said that he was acting in his official capacity. His legal team did not respond to requests from The New York Times to provide evidence to suggest that the committee was conducting an official inquiry.

Judge May has rejected arguments that he was acting solely in his Senate capacity, and said that by Mr. Graham’s own account of the Raffensperger call, he was trying to influence state election rules rather than solely explore federal remedies. Indeed, in a television interview a few days after the call, Mr. Graham said he suggested to Mr. Raffensperger new state procedures for verifying signatures on mail-in ballots and creating an appeal process. (He also said at the time that he made similar calls to Doug Ducey, the Republican governor of Arizona, another state Mr. Trump narrowly lost.)

Mr. Raffensperger’s account of his conversation with Mr. Graham — and his inference that Mr. Graham wanted to explore tossing mail-in votes from counties with high rates of questionable signatures — has been backed up by one of the secretary of state’s aides who was also on the call. Even so, Mr. Graham made no overt request to discard ballots, according to another Raffensperger aide, Gabriel Sterling. Mr. Graham has said that it is “ridiculous” to suggest he was asking for votes to be thrown out.

During a hearing in federal court this month, Brian C. Lea, one of Mr. Graham’s lawyers, said: “We have one phone call, and that phone call has been described by everybody. Everybody acknowledges that it is about electoral process and about verification of absentee ballots, how you ensure security.”

He said the “only dispute” was brought about by Mr. Raffensperger’s account that it was implied that legal ballots should be thrown out. “Strip away the implication that Secretary of State Raffensperger claims to have picked up, all you have is a conversation about electoral process.” Legal precedent, he argued, meant that “motive is irrelevant.”

But Judge May told Mr. Graham’s lawyers that it was critical to understand why the call was made.

“You keep saying that it’s improper for the court to look at motive, but how can I classify an act as political or legislative without knowing why the act was done?”

Knowing what happened around the call could prove critical to Ms. Willis, the Fulton County district attorney.

“The judge in her decision, and the D.A. in her filing, have outlined several different topics beyond the content of the call that could influence the D.A.’s charging decision,” said Gwendolyn Keyes Fleming, a former district attorney of neighboring DeKalb County. “This includes whether there is sufficient evidence of a connection between anything Senator Graham did or said and the former president’s allies — or even the former president himself — to establish some of the elements of a possible conspiracy or RICO charge.”

Ms. Fleming is a co-author on a 114-page Brookings Institution report on the case that found that Mr. Trump himself was “at substantial risk of possible state charges predicated on multiple crimes.” The report called the suggestion that all mail-in ballots from certain counties be disqualified “extreme and bizarre.”

Soon after the call, three legal experts, including Walter M. Shaub Jr., former director of the U.S. Office of Government Ethics, wrote to the chair and vice chair of the Senate ethics committee, decrying Mr. Graham’s contact with Mr. Raffensperger, which took place as Georgia elections officials were conducting a hand recount.

“Any call by a sitting chairman of the Senate Judiciary Committee to a state election official during an ongoing count of votes is inherently coercive and points to an attempt to influence the outcome,” they wrote. “The allegation that Senator Graham placed a behind-the-scenes call to a member of his own political party, without having launched a formal investigation, suggests that he hoped to act out of public view.”

The two other signatories of the complaint, the law professors Claire O. Finkelstein of the University of Pennsylvania and Richard W. Painter of the University of Minnesota said on Wednesday that the ethics committee had not been in touch with them or Mr. Shaub about the complaint, other than to acknowledge that they had received it. The ethics committee staff did not return calls for comment.

Michael J. Moore, a former U.S. attorney in the state, said Mr. Graham “should be afraid of being wrapped up in any conspiracy indictment.”

“I wouldn’t even want to show up as an unindicted co-conspirator in the case she is trying to build,” he added, referring to Ms. Willis. “I don’t know that her RICO efforts will survive appeals,” he said, but Mr. Graham “still wouldn’t want to be in it.”

For Mr. Graham, the case represents the latest chapter in his head-spinning relationship with Mr. Trump. First he was a critic, then a fierce advocate, particularly after the election. But on Jan. 6, 2021, as blood and broken glass were still being cleaned from the U.S. Capitol after the riot by a pro-Trump mob, he declared on the Senate floor, “Count me out” and “Enough is enough.” Two days later, he was jeered as a traitor by Trump supporters at Ronald Reagan National Airport in Washington. Within weeks, he was back on Mr. Trump’s golf course at Mar-a-Lago.