The incident took place as Poyet was sent to the stand in the 35th minute of the Black Cats’ encounter in East Yorkshire.
Bruce was formally reminded of his responsibilities by the FA, while Poyet has requested a personal hearing with the governing body.
The Uruguayan was ordered to the stand by referee Mike Dean for reacting badly to a yellow card for midfielder Jack Rodwell, who was booked for diving.
The Black Cats chief kicked over a water bottle carrier in the dugout and then confronted Bruce, who had to be restrained by an assistant referee and his coaching staff.
After the game, Bruce admitted behaving like a “raging idiot” for his part, while Poyet declared himself “old fashioned” and preferred to leave the matter on the pitch.
The pair shook hands in a KC Stadium corridor after the game, with Bruce, speaking at the time, adding: “We all do stupid things and we will wake up to stupid headlines.
“We’ll both regret it in the morning, but we get on with our lives.
“These things happen in the heat of the moment, grown men acting like a couple of children.
“What he said will stay between me and him...it’s up to Gus to reveal what he said but I don’t think he will.
“It wasn’t very pleasant, like ‘have a nice evening Steve’. But I’m not going to reveal it to sell your newspapers.”
Asked to offer an account of the incident with Bruce, Poyet was less forthcoming last Tuesday night, saying: “No, because I am old-fashioned. What happened stays on the pitch.
“I’ve spoken to Steve and shaken hands. I just saw him, nothing happened. It’s football. It’s passion.
“I have to convince the officials not to put any buckets of drinks around me.
“I have no regrets whatsoever. If that is a bad image for football, I am sorry, but I don’t think it’s something to worry about.”