The Boy Scouts of America said in its statement that when it received information from the plaintiff’s lawyers about the 350 newly identified possible abusers, it began an investigation. It has sent about 120 reports to law enforcement agencies as a result, the organization said, and was told that additional information about some of the cases was needed to follow up on the allegations.

The statement noted that the Boy Scouts had put safeguards in place over the years, including criminal background checks of volunteers; mandatory training for volunteers, parents and scouts; and a policy that prohibits “one-on-one situations” between children and adults.

“We care deeply about all victims of abuse, and sincerely apologize to anyone who was harmed during their time in scouting,” the statement said. “We believe victims, we support them, we pay for counseling by a provider of their choice, and we encourage them to come forward.”

Some of the Boy Scouts’ blacklist files were made public in 2012 as part of a civil trial in Oregon, where the jury awarded $18.5 million in punitive damages to a former scout who said he was abused by an assistant troop leader. At that time, the files contained accusations that had been made against 1,247 scout leaders between 1965 and 1985.

Mr. Eisenberg said in an interview on Wednesday that some of the new allegations involve much more recent abuse. One of the 800 clients represented by Abused in Scouting, he said, is a 14-year-old boy who says he was abused last year.

Abused in Scouting and other lawyers who represent former scouts in abuse cases have been actively encouraging victims to come forward. The group has set up a website and run television ads for that purpose.

“We thought it was important for people to know that if they were sexually abused, they should come forward now,” Mr. Eisenberg said. “If the Boy Scouts declare bankruptcy, it could affect everything.”

In its statement, the Boy Scouts of America said it was “exploring options” regarding a financial restructuring, so it “can live up to our social and moral responsibility to fairly compensate victims who suffered abuse during their time in scouting, while also ensuring that we carry out our mission to serve youth, families and local communities.”