The group, who are in their fourth incarnation, also admitted that Keisha Buchanan's departure in 2009 was "a horrible time", comparing it to a break-up.
"It was a bit of a blur. We got loads of crap as well, and a few times we were like, 'Right, that's it, we've got to tell the world what happened'," Amelle Berrabah told Gaydar Radio.
"We just looked so bad, and [the press] were writing more and more."
Heidi Range added: "It was like going through a break-up... where you know that person's not right for you but you still love them because of the history you've got. Things were getting written saying we'd kicked Keisha out - it was just a really bad time.
"We were trying to be respectful for her and didn't want to air people's dirty laundry in public, and because we didn't talk about the ins and outs, people think we've got something to be guilty of."
Former Sugababe Mutya Buena, who left the band in 2005, was recently revealed to have won minor rights to the Sugababes name.
The Sugababes release their new single 'Freedom' on September 25. Watch the official music video below: