Cellino, 57, is currently waiting for the outcome of Monday’s appeal hearing in London after being disqualified by the Football League from completing a £25million takeover.
The Italian businessman was highly critical of McDermott and club owner Gulf Finance House Capital, but was unaware his conversation on Sunday night was being recorded after taking a cold call from a Leeds fan hosting White Leeds Radio.
McDermott, speaking at a press conference ahead of Tuesday night’s home game against Charlton, said he had listened to the interview.
“If you have a private conversation, it’s private and shouldn’t have gone out, full stop,” McDermott said.
“It was private. What we need is a togetherness here and we’ll get a togetherness.
“If he gets the club he’ll have a decision to make on where we go forward.”
McDermott, 52, has repeatedly stated he will not walk away from the club and would be willing to work with Cellino if his takeover was successful.