Although Real have dominated most of the time and continued attempts to score goals, they have failed to break the Bilbao defense to get a point.
Real finished the game with 10 men after Sergio Ramos was sent off in the 86th minute.
Real have raised the score to 28 points in fourth place with a goal difference against Sevilla, and Bilbao have increased to 14 points in 15th place on goal difference against Ibar.
The Real have been able to take advantage of Barcelona's stalemate against Celta Vigo after a 2-2 draw earlier in the day and a win to cut the gap to six points, but could not make it eight points.
The draw is the fourth for Real this season in return for eight games and two games lost, while the draw is the fifth for Bilbao against a three-game win and a loss in six.
The first half came in the middle of the game and the two sides controlled the game, although the ball was slightly tilted to the Real, who launched successive attacks against Bilbao, whose players initially retreated to absorb the players' enthusiasm.
In the eighth minute, the riyal broke its offensive teeth when Isco played a cross from the left, Karim Benzema received a great ball but his ball hit the bottom of the left post.
Athletic Bilbao's response came in the 13th minute when a cross from the right-hand side was played by Artiz Adores with a powerful header, but Killor Navas shone.
The game was confined to the center of the field until the 28th minute, which saw an offensive awakening for Bilbao, where Ienaki Williams received the ball inside the penalty area on the right and sent a strong ball but hit the foot of a Real defender and went out of the corner untapped.
A minute later, Bilbao missed a solid goal in the 29th minute when she played the ball from a free kick inside the penalty area, which Raul Garcia passed in the six yards to Adoriz, who did not hit the ball to miss a certain goal.
Real came back a minute later when Danny Carvagal broke through on the right and sped past the far-off defense to get in front of Tony Crosse on the edge of the area to give the ball a powerful shot past the right-hand guard Kiba Arizabalaga.
In the 36th minute Ronaldo missed a certain goal when Marcelo played a left-foot ball from Ronaldo's unmarked side inside the Bilbao penalty area on the right-hand side and shot hard but his shot hit the outside net.
The remaining time of the game went through intensive attempts from both teams, but to no avail, the referee fired the whistle of the end of the first half draw a negative between the two teams.
The second half did not change as the attacking teams continued to look for a goal.
Bilbao almost opened the scoring in the 47th minute when Marquel Susaita played a cross from the left and reached Adoriz, who met with a powerful shot but Navas shone to the ball.
In the 55th minute Bilbao resumed his attack when the ball reached Marcell Susiata in the penalty area on the right, where he tried to play from above Navas, but the latter shone and pushed the ball to kick an unused kick.
After the attack, the Real team took control of the game and launched successive attacks on Bilbao. At the same time, Bilbao relied on counter-attacks, exploiting areas in the Real defense, but both failed to form the desired danger on the goal to limit playing in midfield.
The game remained confined to the center of the canned until the 72nd minute, which saw the most serious attacks of the meeting when Ronaldo received the ball inside the penalty area Bilbao and hit a strong ball hit the bottom of the left post.
Real Madrid's attempts to score the winner continued, but the Bilbao players were frustrated and managed to spoil all the Real's attempts.
Ronaldo almost opened the scoring in the 85th minute when Marcelo passed a cross from the left side into the Bilbao penalty area, where Ronaldo shot her up with a header but easily reached the keeper.
The Real received a powerful blow in the 86th minute after Sergio Ramos received the second yellow card and then red after a violent intervention with Bilbao's Adoriz in one of the joint airs.
Bilbao took advantage of the numerical shortage in the Real and started successive attacks in an attempt to score the winning goal, but the Real players were caught in the defense of their own goal to give the referee the whistle of the end of the match, leaving a negative draw between the two teams.