According to The Verge, the threat was made in a letter sent by Sony lawyer David Boies to Vijaya Gadde, Twitter's general counsel, stating that the company would "hold Twitter responsible for any damage or loss arising" from the use of the stolen information.
According to Boies, his client "does not consent to Twitter's or any Twitter account holder's possession, review, copying, dissemination, publication, uploading, downloading or making any use of the Stolen Information".
"If Twitter does not comply with this request and the stolen information continues to be disseminated by Twitter in any manner, SPE will have no choice but to hold Twitter responsible for any damage or loss arising from such use or dissemination by Twitter," the letter read.
Sony specifically complained about Val Broeksmit, a musician who has been tweeting screen grabs of the text of Sony emails on his Twitter account at @BikiniRobotArmy.
It has asked Twitter to suspend Broeksmit's account and destroy any copy of the emails that may be stored on its systems, the report added.
Guardians of Peace threatened top level Sony executives that "We want everything related to the movie, including its trailers, as well as its full version down from any website hosting them immediately", tmz.com reported.
Sony Pictures has suffered an estimated loss of $200 million (roughly Rs. 1,265 crores) on account of the recent cyber-attack over the film "The Interview", according to various analysts.
The film revolves around a fictitious US plot to assassinate North Korean leader Kim Jong-un.
In November this year, the company went through one of the most devastating hacking attacks in corporate history.