The 51-year-old admitted he had his concerns about being replaced when former owner John Majdeski sold the club to the Russian earlier this year.
However, when the two met and McDermott offered to stand down if Zingarevich wanted a new manager, he was categorically told he was the man to lead the club back to the Premier League.
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"When I first met Anton it was in January. He was very interested in buying the club," McDermott told reporters.
"They did a deal in Milan - the chairman and Anton.
"I stayed there that night and we had a conversation. He [Zingarevich] obviously he had his own ideas and opinions.
"I had to get to know him quite quickly, which was interesting in itself. He didn't know me particularly well. He knew me loosely and likewise with the players.
"People often say, 'New owner, new manager' - that generally is what happens.
"I didn't worry. I said to him, 'You make your choice. If you want to find yourself a new manager no problem.' He said, 'No, I want you.' and I said, 'Look, let's see what happens over time. I have no problem whatever choice you make. You make that choice.'
"We get on well. We speak a lot and the way we work has evolved. We work so well together."
McDermott has praised Zingarevich for his relaxed approach which has allowed him to concentrate fully on his job but admitted he takes a deep interest in the game and the club itself.
He added: "Sir John picked some fantastic managers - Steve Coppell, Brendan Rodgers and Alan Pardew - and let them get on with the job. Anton is the same.
"He loves his football. He watches more than anyone I know - even more than me! And it has been really interesting working with him.
"It has been different, but in a good way. He wants to know what's in my head - what I'm thinking. And that's reasonable. I have no problem with that.
"He speaks to Nick [Hammond, Reading's director of football] most days and obviously I work really closely with Nick. And he is kept in the loop. And rightly so."