The Royals are hoping to climb out of their relegation zone when they travel to Mark Hughes' basement club QPR on Sunday.
West Ham boss Sam Allardyce has identified McDermott as one of the managers currently under pressure, but the Reading manager is confident he has the support of Zingarevich.
"The owner was in a team meeting with us last week so he knows what we're about. He has known us since January now," McDermott said.
"Obviously we want more points on the board, but he knows that everybody is giving everything they've got, so he's been really supportive.
"In January he said to me, 'look we don't need to go up now, maybe have a real go at it the following season'.
"I said to him 'I'm 50 years of age now, I need to go up'. So I was the one who wasn't patient, but he was patient. He saw this season maybe a year in advance but now we're here."
Reading have been coming to terms with Tuesday's stunning 7-5 Capital One Cup defeat by Arsenal, a match they led 4-0 late in the first half.
While the result was hard to take, McDermott insists the pain of their npower Championship play-off final defeat by Swansea in May 2011 was harder to take.
"Directly after the game I was in a little bit of shock," McDermott said.
"I drove home, watched the game back and went to bed about 4am. I was up the next day and got on with it because it is a game of football.
"I had a game in the play-off final and after that nothing will ever, ever hurt me again. Nothing."
"Because that as bad as it ever gets. You really have to be proud with some of the stuff that we produced the other night and that's what I'm holding on to."