0458 GMT January 19, 2019
Real could have moved to within six points of Barcelona after the leader was held at home by Celta Vigo earlier on Saturday, but Zidane's men drew a blank in the Basque country, PA Sport reported.
Cristiano Ronaldo and Karim Benzema, who have scored just four goals between them in La Liga this season, again failed to net as Real was forced to settle for its fourth draw of the campaign.
Zidane insisted he is not worried by his side's struggles in front of goal, but conceded that was the only thing that prevented Real from claiming what he felt would have been a deserved victory.
He said in quotes reported by AS, "I'm very happy with the performance and very disappointed not to have scored. But above all for the players.
"I'm very disappointed because with the game we played.... We knew that we had the chance to cut the gap, but the shame is that we did everything but in the end couldn't score. We deserved more.
"We played very well, but the ball didn't want to go in.
"If you score a goal, you cut the gap (to Barca). We can play worse, and we have done, but today and against Atletico (another 0-0 draw) we deserved much more than the opponent. With goals you have four points more and it's different."
Asked if he was concerned about the lack of goals, Zidane said, "I'm not worried, I'm disappointed for the players.
"They are disappointed. My job is (to) think positively. Playing in this fashion the goals are going to come."
Zidane selected the same XI that beat Juventus in June's Champions League final for the first time since that match, but whereas Real won that game 4-1, Los Blancos were forced to settle for a 0-0 draw on Saturday.
On what was different on this occasion, the Frenchman added, "The result, nothing more. When you look at today's game you cannot say we played worse. I don't think so.
"What's changed is the goal. Against Juve we scored four and today none."
After failing to score, the champion was left holding on for even a point at the end after captain Sergio Ramos was sent off – the 24th time he has seen red in his Real career.
Zidane said of the Spain center-back: "He's not happy because he thought it wasn't a card. But that's that. We cannot change it now."