The United States Court of Appeals for the Fifth Circuit, in New Orleans, said Sergio’s family could not sue Mr. Mesa.

“This is not a close case,” Judge Edith Jones wrote for the majority in March. Congress could pass a law allowing suits against federal officials by “aliens injured abroad,” she said. But without such a law, she wrote, federal courts should not “interfere with the political branches’ oversight of national security and foreign affairs.”

The Supreme Court will soon consider whether to hear the Texas case, Hernandez v. Mesa, which has been before the justices once before. In the Arizona case, Rodriguez v. Swartz, a lawyer for Mr. Swartz filed papers last week indicating that he would seek Supreme Court review.

Image Judge Andrew J. Kleinfeld of the United States Court of Appeals for the Ninth Circuit in 2007. He wrote that José Antonio’s mother should not need congressional permission to sue. Credit... Al Grillo/Associated Press

If the Supreme Court agrees to hear one or both of the cases, the families of the boys may face an uphill fight. And there is little reason to think that Judge Brett M. Kavanaugh, President Trump’s Supreme Court nominee, would be sympathetic to their arguments.

In 2015, Judge Kavanaugh, who sits on the United States Court of Appeals for the District of Columbia Circuit, joined a decision dismissing a lawsuit filed by an American citizen who said he had suffered months of abuse at the hands of American law enforcement officials in Kenya, Somalia and Ethiopia. In a concurring opinion, Judge Kavanaugh said the plaintiff, Amir Meshal, could not sue the officials for violating his constitutional rights because no statute authorized such suits.

In 1971, in Bivens v. Six Unknown Named Agents, the Supreme Court ruled that such congressional authorization was not always needed for people to sue federal officials for violating their constitutional rights. But the court has grown increasingly uneasy about the decision, which concerned the unconstitutional search of a home in Brooklyn, and it has cautioned that the decision should not lightly be extended to new contexts. The Supreme Court itself has extended the Bivens decision only twice, most recently 38 years ago.