NEW YORK — Microsoft Xbox Live and Sony PlayStation Network, the Internet services video gamers use to play online, were hit by connection failures on Christmas Day, with the hackers Lizard Squad claiming responsibility.

The group, which took credit for an attack on Sony earlier this year, said on its Twitter account that it was behind the attacks.

“I have the nation on strings,” Lizard Squad wrote on its Twitter site.

The group said it would “stop hitting” the services if users called attention to the hack by retweeting its statements.

Thursday’s incidents are part of a series of escalating attacks against Sony. This month, a group called Guardians of Peace claimed responsibility for hacking into the Tokyo-based company’s servers, destroying data, exposing Hollywood secrets, and forcing Sony’s movie studio to cancel the nationwide theater release of “The Interview,” a comedy about a fictional assassination of North Korean leader Kim Jong Un. Instead, Sony released the movie online and on the Xbox network. Both hacking groups had threatened further disruptions on Christmas Day.

Users were having difficulty signing into their online accounts and Xbox was working to fix the problem, according to an alert on Xbox Live’s website. The PlayStation Network, used to connect Sony’s gaming consoles, was experiencing similar issues.

“We are aware that some users are experiencing difficulty logging into the PSN,” Sony Computer Entertainment America said on its website. “We will update this article with any changes that occur in regards to this issue.”

An email and voicemail to David Dennis, a spokesman for Microsoft, weren’t immediately returned. Messages to Jennifer Clark and Sean Yoneda, representatives for Sony, weren’t immediately returned.

Last week, President Barack Obama blamed North Korea for orchestrating the attacks against Sony and vowed to respond. North Korea has said it doesn’t know the identity of the hackers claiming responsibility for breaking into Sony’s computer network. The country’s connection to the Internet was also disrupted earlier this week.

Sony Pictures released “The Interview” online Wednesday via Google’s YouTube video-streaming website, the Xbox video console and www.seetheinterview.com, a website sponsored by Sony. By streaming the comedy via the Web, Microsoft and Sony took the risk of provoking denial-of-service hacking attacks. The hacktivists had warned that they intended to target the companies with such incidents on Christmas Day.

Denial-of-service assaults can be difficult to deflect, even if a company has ample warning they are coming, because they are executed by thousands of hacked computers performing normal but database-intensive activities, such as performing searches or downloading videos, all at the same time.