Thirteen female athletes who were sexually assaulted by Lawrence G. Nassar, the former sports doctor for the U.S.A. Gymnastics national team and Michigan State University, are seeking $10 million each from the F.B.I., alleging that its agents mishandled an investigation and allowed Mr. Nassar to continue abusing more victims, a lawyer said on Thursday.

The lawyer, Jamie White, said he filed a tort claim on Wednesday against the bureau that gives the government six months to settle or deny the claim. A lawsuit could follow, depending on the response, he said.

It was the latest legal action to arise from the abuse of young athletes by Mr. Nassar, who is serving what amounts to life in prison for multiple sex crimes involving girls and women, including members of the 2012 and 2016 U.S. Olympic women’s gymnastics teams.

Last year, the Justice Department’s inspector general issued a report that sharply criticized the F.B.I.’s handling of the case, saying that Mr. Nassar had abused 70 or more young athletes between July 2015, when U.S.A. Gymnastics first reported allegations against him to the F.B.I.’s Indianapolis field office, and August 2016, when the Michigan State University Police Department received a separate complaint.

The 13 athletes who are seeking payments from the F.B.I. were abused by Mr. Nassar when they were gymnasts at the Twistars gymnastics club, outside Lansing, Mich., Mr. White said in an interview. John Geddert, who owned Twistars and referred injured athletes to Mr. Nassar, fueling a cycle of abuse, killed himself in 2021 after he was charged with human trafficking, including forced labor, and sexually assaulting a teenage girl.

The athletes, who were middle school or high school students at the time, were assaulted after the F.B.I. was first made aware of the abuse allegations in 2015, Mr. White said. Some were assaulted before, as well, he said.

“Some were assaulted before 2015, and all were at least assaulted after,” he said. Some received physical and mental health treatment as a result of the abuse, and one was admitted to a treatment facility last week, he said.

The filing document said the F.B.I. field offices in Indianapolis and Los Angeles mishandled reports of Mr. Nassar’s abuse. It said the Indianapolis office learned about the abuse from U.S.A. Gymnastics in July 2015 but “failed to formally document” the information and evidence it gathered, including through interviews with gymnasts.

The Los Angeles field office learned of the abuse in May 2016 and “failed to notify local law enforcement,” the document said.

Those failures left Mr. Nassar “free to commit unspeakable sexual assaults on many victims,” many of whom were children at the time, the court document said.

One of the victims, the document said, was sent to Mr. Nassar for treatment for an injury and pain, hoping to be able to resume her training in gymnastics and ballet. She was sexually assaulted on 17 to 20 occasions, the document said. She and her parents did not know that the F.B.I. office in Indianapolis was aware that Mr. Nassar had been accused of sexual abuse, the claim said.

Mr. White said his approach was similar to one that led the Justice Department to pay about $130 million last year to 40 survivors and families of victims of the 2018 massacre at a high school in Parkland, Fla. In that case, the F.B.I. acknowledged that it had failed to properly investigate two tips that suggested the gunman might open fire at a school.

The Justice Department has sharply criticized the F.B.I.’s handling of the Nassar case. Last year, the F.B.I. director, Christopher A. Wray, who took charge in 2017, acknowledged the agency’s mishandling of the case. He told the Senate Judiciary Committee that the bureau had strengthened its policies and training, promising that steps in future investigations would be “quadruple checked” so that there would not be “a single point of failure.”

“I’m sorry that so many people let you down again and again,” he said, addressing some of the athletes whom Mr. Nassar had abused and who testified at a Senate hearing. “I am especially sorry that there were people at the F.B.I. who had their own chance to stop this monster back in 2015 and failed, and that is inexcusable. It never should have happened, and we are doing everything in our power to make sure it never happens again.”