The Spanish giants have won just four of their last 10 matches in all competitions, with Saturday's 1-0 defeat at Mallorca heaping the pressure on Zidane.
Madrid are also bottom of Group A in the Champions League, following a 3-0 loss at PSG with a 2-2 home draw against Club Brugge last time out.
Speculation that Mourinho could return to the Santiago Bernabeu has raged in recent days and was a hot topic during Madrid's news conference on Monday, just over 24 hours before a crucial Champions League fixture at Galatasaray.
When told defender Sergio Ramos had just defended him in light of the Mourinho talk, Zidane told reporters: "In football, people forget what you've done in the past.
"The important thing is the present…I'm not going to say it doesn't bother me what is being said about other coaches, because it does.
"But what I have to do as a coach is to give everything, get the players to do the same, to give everything – and that's all I can do.
"It's a great game to play in, a great stadium, Champions League…we know what the situation is and we're going to try and win."
Asked if he felt he could lose his job if Madrid are defeated again, Zidane said: "I don't know. You can't ask me that. All I can do is try and be here as long as I can. I want to be here forever."
Zidane added that pressure is part of being at Madrid and that his team are prepared to respond.
"We know what the situation is," he said.
"We started badly in the Champions [League] but we've got a chance tomorrow to prove ourselves and that's what we are going to try and do … I've been here 18 years as a player and a coach.
"I know how things are. Pressure is always with you so we have to go out and prepare well for the game."
Ramos said that the players are only thinking about winning in Istanbul.
"We're not looking at opinions from the outside," he said.
"We are quite calm, inwardly. In the dressing room, we know we need to work. We just need to concentrate on our games and our plans."