“The game was lost in the first half when Athletic scored and then in the second half we were unable to get back into the match,” Real coach Carlo Ancelotti told a news conference.
“It is not a physical problem as in the second half we had a lot of the ball but we were unable to create clear chances.”
Real have taken just one point from their last two games and, although they are two points clear, will be overhauled if Barcelona beat visitors Rayo Vallecano on Sunday.
“The league is difficult now but it was also difficult when we had a four-point lead,” Real midfielder Isco told reporters.
“There is a long way to go and there are a lot of games. Things didn’t go well today but they can change from one moment to another. Our heads are a bit down now but that can change quickly.”
Real were denied time on the ball by Bilbao, who are on a high after booking a place in the King’s Cup final with victory over Espanyol on Wednesday.
Bilbao were unable to maintain the pressure after the break and Real began to dominate possession but the visitors struggled to create openings.
Instead Aduriz came within inches of getting a second goal with another header which hit the outside of the post.
Real forward Gareth Bale almost equalised but hit the post with a spectacular effort seven minutes from time when he lobbed keeper Gorka Iraizoz midway inside the Bilbao half.
DRAMATIC VICTORY
Sevilla's Vitolo scored twice as they put their Champions League qualification hopes back on track with a dramatic 4-3 victory at Deportivo La Coruna.
Another double from Depor's Oriol Riera could not prevent Sevilla picking up their first win in three league games, which cements fifth place in the table on 49 points.
Riera opened the scoring in the 28th minute only for Vitolo to quickly respond with an equaliser.
The Sevilla striker then demonstrated more clinical finishing after 52 minutes, having been allowed space in the centre of the penalty area.
Sevilla's Kevin Gameiro converted a controversial penalty after Vitolo, played onside by injured defender Juanfran, was felled by Pablo Insua in the box.
Riera pulled a goal back for Deportivo to make it 3-2 before their Brazilian central defender Sidnei diverted a Vitolo shot into his own net.
As the conclusion to a contest featuring some hapless defending, Sevilla defender Nicolas Pareja upended Helder Costa for another spot kick, which Depor's Lucas Perez struck home.
Elche beat Almeria 1-0 after Victor Rodriguez hit a sixth-minute winner while second-bottom Granada won by that scoreline against Malaga with a 57th minute Ignacio Camacho own goal.
Elche's win has taken them out of the relegation places and onto 27 points while Almeria are now third-bottom with 24, one point below the safety zone and two points above Granada.
RARE START
In Bilbao, Real's Asier Illarramendi was given a rare starting place in the midfield where they are still missing the injured Luka Modric and James Rodriguez.
The hosts were missing highly-rated defender Aymeric Laporte so Carlos Gurpegi started alongside Xabier Etxeita at the back.
Real were unsettled by a high-tempo start from Bilbao and were unable to link up their play in the middle of the pitch.
Their frustration was clearly shown by a wild kick from Toni Kroos on Andoni Iraola, shortly after Aduriz’s goal, that brought the Germany midfielder a yellow card.
Exciting Bilbao youngster Inaki Williams and Iker Muniain stretched the Real defence and when the visitors did look to attack they appeared tentative.
Bilbao could not keep up the same rhythm throughout the match but Real were unable to take advantage.
Ancelotti's side have lost their way with just two victories in five games and they misfired again in attack.
They were reduced to shots from distance and in the final stages, aside from Bale's effort, Bilbao seemed comfortable.
(Editing by Ken Ferris)
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