Cellino's protracted takeover of the Elland Road outfit was finally given the approval of the Football League last week after they accepted the verdict of an independent QC to overturn their initial rejection.
The uncertainty over the ownership of the West Yorkshire side has coincided with a miserable run of form which has seen them lose eight of their last 10 games to ramp up the pressure on McDermott.
But the former Reading boss, who was sacked in January by a lawyer - believed to be acting on Cellino's instruction - before he was then reinstated a day later, saw his side defeat Blackpool 2-0 at home on Saturday.
And with the club now on a more stable footing, after a spell where the management and players did not receive their full wages, McDermott believes he is the right man to take them forward under Cellino.
"He's his own man," McDermott told the Yorkshire Evening Post. "He'll do what he thinks is right, and rightly so. He's the owner of the club.
"But I've won this league. I've won it in a stable environment. You're not talking about a manager who hasn't.
"I've also been in a play-off final and I had a 50 per cent win ratio in this league (with Reading) so I know what's required.
"I haven't got that at Leeds but it's where we have to get to. Given an opportunity, I feel so strongly that I can do it."
McDermott insists he never considered turning his back on Leeds in January, when he was axed and then almost immediately reinstated in the early stages of Cellino's bid to take control.
"I could have walked away in January but the thought never even entered my head," he said. "I've been to some tough places but it never entered my head.
"It's got nothing to do with money. I'm manager of Leeds United. Why would I want to leave or even consider leaving a job like that?
"Where am I going to go from Leeds United if I walk away? That's the thing. It means too much to me."