The man was attacked in the Bessemer pub in Leopold Square at around 6pm on Thursday June 16.
South Yorkshire Police said a group of men were involved in the incident, which left the victim with wounds to his neck and a slash wound to his ear, which required hospital treatment.
PC Leanne Bennett said: “We understand the incident happened when two off-duty members of bar staff trying to eject a group of people from the pub that evening.
“It is believed that the victim’s injuries were caused by broken glass and could have been much more severe.
“It’s important that if you were in the pub that evening and witnessed the incident that you get in touch so we can identify those involved.”
Anyone with information should call South Yorkshire Police on 101 or Crimestoppers on 0800 555111.