The chairman of the CHUM's board, Jean-Claude Deschênes, also announced Friday at a news conference he was resigning over the matter.
Turgeon said Barrette pressured him to select a certain candidate as the CHUM's chief of surgery.
He said Barrette was interfering in the governance of the hospital, adding that the best thing for him and the institution was for him to step down.
When asked if he thought Barrette should remain as health minister, Turgeon said "that's a decision for the premier."
Barrette told CBC Friday that he never interfered in staffing decisions and that he only made "suggestions."
The health minister has insisted Turgeon's resignation will not lead to delays in the CHUM superhospital project.