The group, called Lizard Squad, could not be reached for comment, and the companies did not give a reason for the connectivity issues.
Both companies expected heavy use as people who received Microsoft Xbox players or Sony PlayStations on Christmas tried to hook up. The problems may have been exacerbated by Sony's decision to let consumers download the controversial movie "The Interview" through Xbox Video as well as other online sources.
"Are you having a rough time signing in to Xbox Live?," a service alert on the Microsoft product's website asked. "We're working to get this figured out right away. We appreciate your patience."
Two hours after the message was posted at 4:33 p.m. ET (21:33 GMT), the Xbox Live site said technicians were still "hard at work trying to reach a solution."
As of 7:30 p.m. ET (0030 GMT), PlayStation Network was still offline, according to the device's status page.
"We are aware that some users are experiencing difficulty logging into the PSN," it said. "We will update this article with any changes that occur in regards to this issue. Thank you for your patience."
"I have the nation on strings," Lizard Squad wrote on its Twitter site. It also said it would put the sites back online if enough people retweeted its messages.
Earlier on Thursday, Microsoft Xbox spokesman Sean McCarthy declined to give details of how the company may have been preparing for security breaches, given the problems that Sony has experienced in releasing "The Interview."
"Of course, it's safe to say Holiday season is always a very busy time of year for any consumer electronics company, so we work hard to ensure the stability of our infrastructure when so many consoles are activating for the first time."
© Thomson Reuters 2014