Real are in the final Champions League spot, now only one point above Villarreal - who had never won at the Bernabeu.
Considering Madrid also drew 2-2 at home to Numancia in the cup last week, can Zidane survive?
Costa was suspended because of a red card for celebrating too close to the fans in the previous game.
The misfiring champions were beaten by an 87th-minute strike from Pablo Fornals at the end of a superb break by the visitors as Real pressed for a victor.
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In La Liga, Javi Calleja's side are in sixth, just four points behind Madrid, but they will be without midfielder Bruno Soriano, defender Ruben Semedo and reserve goalkeeper Andres Fernandez on Saturday, while forwards Nicola Sansone and Roberto Soldado are doubtful.
Nacho insists that there is nothing wrong with Real Madrid's attitude after Los Blancos' poor form continued with a demoralising 1-0 defeat to Villarreal.
Beaten 1-0 by Villarreal and with their title hopes in total disarray on a rain swept Saturday, Zidane's men trudged off the sodden pitch with jeers ringing around the Santiago Bernabeu Stadium.
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ASH Barty will need to become the first player to defeat Angelique Kerber in 2018 if she is to win the Sydney International. I feel like if I play like that, I'll probably beat most of the people in the draw.
"We have endured negative results recently, but we know that we can change this situation by winning again", Zidane said.
Villarreal have won their last two away league games, both by the same scoreline (1-0).
Florin Andone headed in a late consolation for the home side with three minutes left.
Real - who saw a Cristiano Ronaldo free-kick turned on to the bar - had 28 shots, more than any other side in a La Liga game this season.
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Carlos Bacca has been involved in six of Villarreal's last 10 goals in La Liga (three goals, three assists).
Real controlled much of the first half, and had the first sight of goal on 17 minutes when Marcelo lashed a strike from distance which Sergio Asenjo turned away with a strong fingertip.
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Davis has not said that Gruden verbally accepted the job, but used the phrase "all in" to describe how Gruden felt about the job. The NFL said Thursday it will review the process that led to the Oakland Raiders' hiring of Jon Gruden as their head coach.