Real was beaten 2-0 by Shakhtar Donetsk in the Champions League on Tuesday evening leaving the Spanish giant with the very real threat of playing in the Europa League after Christmas, Mirror reported.
It followed defeat by Alaves in La Liga at the weekend, a result which left the club fourth in the Spanish top flight, seven points behind pace-setter Real Sociedad.
And, according to Marca, Zidane now has just four games to save his job with Pochettino waiting in the wings.
It represents a crunch period for Zidane and Real, with a Madrid derby against Atletico and El Clasico against Barcelona.
Real also takes on Sevilla and Borussia Monchengladbach in the Champions League as it bids to rescue its season.
World Cup winner Zidane, who led the Bernabeu giant to three successive Champions League crowns in his first spell in charge, insists he will not resign.
He said after the Shakhtar result, "I am not going to resign – not at all. We have always had delicate moments. It is true that it is a bad streak in terms of results, but we must continue.
“It is a difficult moment but I am convinced that we are going to bring out our character. We have to win the next game. I am convinced that we can do it."
But the decision could be taken out of his hands, with Pochettino reportedly the name at the top of Real's list of potential replacements.
It is claimed personal terms are already between the two parties.