"We're aware of this issue, and we're working to find a fix ASAP!" the British website for the Microsoft-made Xbox said online.
"We are still looking into the (PlayStation network) issues," the Sony-owned makers of the console tweeted from the AskPlayStationhandle.
Neither Microsoft nor Sony commented on the hacking claim made on Twitter by a user or users under the name Lizard Squad, who said the online attacks would continue until more Twitter users followed the feed.
A British 10-year-old said it was "his worst Christmas ever," after being unable to download games to play on the Xbox he received as a gift, his mother Ros Bruce told the BBC.
"He has spent most of the day in tears," she said. "Xbox should compensate us all."