Former Calgary Flames GM Craig Button has also been mentioned as a replacement for Blue Jackets GM Scott Howson.
However, Priest vehemently denied these claims, indicating that changes of this magnitude would be spearheaded by majority owner John P. McConnell.
"Let me be clear here, though," Priest told the Dispatch. "Everybody understands that the losing can't continue."
Hitchcock, who led the Jackets to their only playoff appearance in 2008-09 and was let go of his duties in Feb. 2010, has also denied the reports, while Button publicly dismissed the notion via his twitter account.
"While it's flattering to be mentioned in GM rumors, I've had no contact with Columbus," Button wrote. "I wish Scott H. the best to turn things around."
The Blue Jackets sit last in the NHL with a 2-9-1 record. The registered only their second win of the season against the Anaheim Ducks on Sunday.
Meanwhile, Howson remains focused on trying to improve his club's chances on the ice and has been fielding calls from his counterparts around the league.
Several members of the Blue Jackets have been linked to trade speculation, but it's unlikely players like Derick Brassard, Marc Methot or Kris Russell will be dealt.
As the chatter continues, the team held a players-only meeting roughly two and a half hours before last night's game against the Ducks.
"The players here love what we've got," captain Rick Nash told the Dispatch. "Management, coaches ... everyone's very excited and very happy with what we have. And we wouldn't want to lose it.
"As players, you feel guilty because you're obviously not winning, that's the reason. But we all believe in our system, in our management. We believe in Arnie. That's the last thing we'd want to happen."