The defending Spanish champions were again frustrated on their travels, as they failed to find a way through in the Basque Country for a fourth draw of a frustrating campaign.
To make matters worse for Los Blancos, they lost captain Sergio Ramos to a red card late on - his 24th for the club - in a match that had a slight edge to it right from the start.
Madrid, beaten in just one of their last 16 league meetings with Athletic, came close to opening the scoring nine minutes in when Karim Benzema chested down Isco's cross and volleyed against the frame of the goal.
Aritz Aduriz guided a header on target for the returning Keylor Navas to keep out soon after, while Kepa Arrizabalaga - heavily linked with a move to Madrid in January - was also needed to block rather tame attempts from Cristiano Ronaldo and Toni Kroos.
Despite being well out of form domestically, Athletic were creating some good openings as Inaki Williams forced Navas into a low save and Raul Garcia somehow failed to pick out the net or a teammate from a glorious back-post position.
In another sign that Ronaldo is still not quite at the races this season - in La Liga, at least - the Portuguese hammered the ball wide from seven yards out from a chance that he has previously made a habit of burying.
The defending champions upped things late in the first half in search of a breakthrough, with Raphael Varane and Ramos both coming close from headers, the latter seeing his attempt blocked by Xabier Etxeita in front of goal.
Aduriz blasted down the middle from the first sight of goal in the second half, before Markel Susaeta picked out the side-netting after his initial lob was saved by Navas.
Madrid, now with just one away win in five in all competitions, continued to struggle when it came to sustaining some pressure as Mikel Rico produced a big block to thwart Isco.
Ronaldo blasted another shot goalwards in the final 20 minutes, skimming the post on its way through, and Kepa was required to keep out a Marcelo drive.
Just as momentum was being built, with Ronaldo guiding another header on target, Ramos - back in the side after three matches on the sidelines - caught Aduriz with a stray arm to be shown a second yellow three minutes from time.
There was to be no late goal for the 10-man visitors, meaning that the gap on Barcelona remains at eight points following the leaders' surprise 2-2 home draw with Celta Vigo earlier today.
Athletic Bilbao: Kepa; De Marcos (Lekue 72'), Etxeita, Nunez, Balenziaga; Iturraspe (San Jose 39'), Mikel Rico; Williams (Cordoba 83'), Raul Garcia, Susaeta; Aduriz
Real Madrid: Keylor Navas; Carvajal, Ramos, Varane, Marcelo; Modric, Casemiro, Kroos; Isco (Kovacic 82'); Benzema (Mayoral 82'), Ronaldo