The incident happened ahead of the tournament semi-final in Portugal.
Purple PR sent the press release announcing Sterling's captaincy on behalf of the player's management agency, Colossal Sports Management.
The England striker admitted he was "fuming" over the leak and told Southgate it was done without his permission or knowledge.
The Telegraph this morning reports that Sterling and his agent, Aidy Ward, are looking at alternative PR arrangements.
Sterling said of the incident at the time: "I had a conversation with Gareth. I hadn't had a conversation with anyone in my agency, so it was a strange one to wake up to.
"The first thing I did after I came off the phone to my agent was see Gareth down the hallway, and I apologised to him for what had happened. I don't know where that's come from.
"[Being captain] that's something that's down to the manager, not for me to decide. But if the manager did do that, it's something I'd be proud and happy to do for sure. That's up to the manager."
Purple declined to comment. PRWeek was unable to contact Colossal.