Justice Samuel A. Alito Jr. received a call on his cellphone Tuesday. It was President-elect Donald J. Trump, calling from Florida.

Hours later, Mr. Trump’s legal team would ask Justice Alito and his eight colleagues on the Supreme Court to block his sentencing in New York for falsifying business records to cover up a hush-money payment to a pornographic film actress before the 2016 election. And the next day, the existence of the call would leak to ABC News — prompting an uproar about Mr. Trump’s talking to a justice before whom he would have business with substantial political and legal consequences.

Justice Alito said in a statement on Wednesday that the pending filing never came up in his conversation with Mr. Trump and that he was not aware, at the time of the call, that the Trump team planned to file it. People familiar with the call confirmed his account.

But the fact of the call and its timing flouted any regard for even the appearance of a conflict of interest at a time when the Supreme Court has come under intense scrutiny over the justices’ refusal to adopt a more rigorous and enforceable ethics code.

The circumstances were extraordinary for another reason: Justice Alito was being drawn into a highly personalized effort by some Trump aides to blackball Republicans deemed insufficiently loyal to Mr. Trump from entering the administration, according to six people with knowledge of the situation, who spoke on the condition of anonymity to describe private conversations.

The phone call centered on William Levi, a former law clerk of Justice Alito’s who seemingly has impeccable conservative legal credentials. But in the eyes of the Trump team, Mr. Levi has a black mark against his name. In the first Trump administration, he served as the chief of staff to Attorney General William P. Barr, who is now viewed as a “traitor” by Mr. Trump for refusing to go along with his efforts to overturn his loss in the 2020 election.

Mr. Levi has been under consideration for several jobs in the new administration, including Pentagon general counsel. He has also been working for the Trump transition on issues related to the Justice Department. But his bid for a permanent position has been stymied by Mr. Trump’s advisers who are vetting personnel for loyalty, according to three of the people with knowledge of the situation.

As Mr. Trump puts together his second administration, Mr. Barr is among a handful of prominent Republicans who are viewed with such suspicion that others associated with them are presumptively not to be given jobs in the administration, according to people familiar with the dynamic. Republicans in that category include Mr. Trump’s former secretary of state Mike Pompeo and his former U.N. ambassador Nikki Haley. To be called a “Pompeo guy” or a “Haley person” is considered a kiss of death in Mr. Trump’s inner circle. Resistance to such people can usually be overcome only if Mr. Trump himself signs off on their hiring.

Tuesday’s phone call took place against that backdrop. Several people close to the Trump transition team on Thursday said their understanding was that Justice Alito had requested the call. But a statement from Justice Alito framed the matter as the justice passively agreeing to take a call at the behest of his former clerk.

The disconnect appeared to stem from Mr. Levi’s role in laying the groundwork for the call in both directions. It was not clear whether someone on the transition team had suggested he propose the call.

Mr. Levi did not respond to a request for comment. The Supreme Court press office said it had nothing to add to the statement it put out from Justice Alito on Wednesday. In that statement, Justice Alito said that Mr. Levi “asked me to take a call from President-elect Trump regarding his qualifications to serve in a government position. I agreed to discuss this matter with President-elect Trump, and he called me yesterday afternoon.”

He added: “We did not discuss the emergency application he filed today, and indeed, I was not even aware at the time of our conversation that such an application would be filed. We also did not discuss any other matter that is pending or might in the future come before the Supreme Court or any past Supreme Court decisions involving the president-elect.”

During the call, according to multiple people briefed on it, Mr. Trump initially seemed confused about why he was talking to Justice Alito, seemingly thinking that he was returning Justice Alito’s call. The justice, two of the people said, told the president-elect that he understood that Mr. Trump wanted to talk about Mr. Levi, and Mr. Trump then got on track and the two discussed him.

A spokesman for Mr. Trump did not respond to an email seeking comment.

While it is unusual for an incoming president to speak with a Supreme Court justice about a job reference, it is routine for justices to serve as references for their former clerks. Justices traditionally treat their clerks as a network of protégés whose continued success they seek to foster as part of their own legacies.

Seemly or not, there is a long history of interactions between presidents and other senior executive branch officials and Supreme Court justices who sometimes will have a say over the fate of administration policies.

In 2004, a controversy arose when there was a lawsuit seeking disclosure of records about Vice President Dick Cheney’s energy task force meetings. One of the litigants, the Sierra Club, asked Justice Antonin Scalia to recuse himself from participation in the case because he had recently gone duck hunting with Mr. Cheney. Justice Scalia declined, issuing a 21-page memorandum that explained why he believed stepping aside was unjustified.

Part of Justice Scalia’s argument was that Mr. Cheney was being sued over an official action. That makes Mr. Trump’s pending attempt to block his sentencing for crimes that he was convicted of committing in his private capacity somewhat different, although the basis of Mr. Trump’s argument is that being sentenced and then fighting an appeal would interfere with his ability to carry out his official duties.

In trying to justify his decision not to recuse, Justice Scalia noted that justices have had personal friendships with presidents going back years, including some who played poker with Presidents Franklin D. Roosevelt and Harry S. Truman but did not recuse themselves from cases challenging their administrations’ policies and actions.

Mr. Trump has long sought to pressure the Supreme Court, in some cases by publicly hectoring the justices on social media for decisions he disagrees with. Mr. Trump has often privately complained that the three justices he appointed in his first term — Neil Gorsuch, Brett Kavanaugh and Amy Coney Barrett — had “done nothing” for him, according to a person who has discussed the matter with Mr. Trump.

One week after the 2018 midterm elections, Mr. Trump and the first lady, Melania Trump, had lunch with Justice Clarence Thomas and his wife, Virginia Thomas. Ms. Thomas, a longtime conservative activist, made suggestions about personnel shake-ups to Mr. Trump and later supported his efforts to try to overturn the 2020 election results.

In December 2020, Mr. Trump attacked the Supreme Court as “incompetent and weak” for refusing to address his legal team’s efforts to challenge the 2020 election. Two years later, he attacked the court again for giving Congress access to his tax returns.

The Supreme Court redeemed itself in Mr. Trump’s eyes last summer when the six Republican-appointed justices ruled that former presidents have broad immunity from being prosecuted over actions they took in their official capacity. That ruling threw into doubt how much of the indictment brought against Mr. Trump for his efforts to overturn the 2020 election could actually survive to go to trial — even after prosecutors filed a revised version trying to account for the court’s decision.

The Supreme Court’s intervention also seriously delayed the case’s progress, effectively making it impossible to get the charges to a jury before the election. And once Trump won the 2024 race, he could no longer face prosecution under Justice Department policy.

Kirsten Noyes contributed research from New York.