At issue is publicity surrounding the case and the concerns some witnesses have that their names and addresses may be published through social media and on the Internet.
The current juvenile court judge overseeing the case scheduled next month in Steubenville has previously declined a request to close the proceedings.
"They are reluctant to sacrifice their college career, their reputation, or their otherwise good standing in whatever community they may be found for fear of being vilified, and certain personal information finding its way on the Internet," said Walter Madison, an attorney for 16-year-old defendant Ma'Lik Richmond.
Ohio Attorney General Mike DeWine, whose office is overseeing the case's prosecution, said the decision will be up to the judge. He declined to comment on the attorneys' concerns.
"There is obviously a lot going on in the social media and there's a lot going on in the mainstream media, but ultimately this case will be decided by a juvenile judge who will listen to the evidence, and the case will be presented no differently than it would in any other case," DeWine said.
Brian Duncan, an attorney for the other 16-year-old defendant, Trent Mays, also said Monday he's considering a similar request.
"We just want to make sure our client and the other defendant have their proper day in court," Duncan said.
The two boys are set for trial next month in juvenile court in Steubenville, a city of about 18,000, on charges that they raped a 16-year-old girl in August. Their attorneys have denied the charges in court.
Public interest in the case increased with circulation online last week of an unverified video, more than 12 minutes long, that purportedly shows another young man joking about the accuser. The video apparently was released by hackers who allege more people were involved and should be held accountable.
Over the weekend, city authorities launched a website to combat misperceptions about the case, including the allegation that the football team has an unusual sway over the city.
The site, sponsored by Steubenville city and police officials, explains that only a handful of police officers attended local schools and that the city manager herself is not even from Ohio. Its launch followed the hiring of a consultant who's helping the city handle a barrage of media attention sparked by the case.
As the investigation continues, it has spurred heated commentary online. Some support the defendants and question the character of the teenage girl, while others allege a cover-up or contend more people should be charged.
The latter group includes hacker-activists associating under the Anonymous and KnightSec labels who point to comments they say were posted around the time of the alleged attack on social media by people who are not charged.