Bullen branded the Owls’ display on New Year’s Day as “embarrassing” and “totally unacceptable” as goals from Tom Flanagan, Lloyd Dyer and Tom Naylor consigned them to a third successive Hillsborough loss.
Defeat leaves Wednesday just six points above the relegation zone.
Bullen said: “The sooner the club appoints a new coach the better. Everybody can move on. As it stands, there’s no announcement or new coach through the door.
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“I feel sorry for the chairman. He came in and shook the hands of the players. The man has done so much for the football club but there have been too many performances this season that have not been acceptable.”
When asked if there has been any development on that score, Bullen said: “None that I am aware of. I’m in constant contact with the chairman. My job is to lift the players mentally and prepare them for Carlisle.”
Supporters booed the team and some chanted ‘You’re not fit to wear the shirt’.
“I can understand the fans’ frustration 100 per cent,” he said. “There was a bloke at the end there shouting to me ‘embarrassing’ and how can I say anything different? He was 100 per cent spot on.
“In the changing room, there were a couple of boys who were vocal enough to dig out their teammates or performance and said the same. The fans have every right after that result to say what they say.
Meanwhile, the Owls have appointed Katrien Meire as their new chief executive officer.
Meire boasts a wealth of experience, having previously worked at Belgium outfit Standard Liege and Charlton Athletic.
Meire, a Football Association council member, stepped down as a director and CEO at Charlton at the end of last year.
She said: “I am so pleased to take up the position of CEO at Sheffield Wednesday. This is a massive challenge at a massive club and I am looking forward to beginning this journey. The chairman has very exciting plans for the club in the short and long term, and I bought into his vision immediately.”