Barcelona threatened to break the deadlock in the first period when Joao Cancelo saw his long-range strike kept out by a combination of Unai Simon and Yeray Alvarez.
Both sides probed for a winner in the second period, but neither team were able to find the breakthrough in a cagey contest.
As a result, Barcelona missed the chance to close the gap to leaders Real Madrid, leaving them in third place and eight points adrift of their Clasico rivals.
Meanwhile, fifth-placed Athletic lose ground in their quest to secure Champions League qualification, with Atletico Madrid five points clear in fourth position.
Athletic began the contest with hopes of ending a memorable week on a high note after they thrashed Atletico Madrid in the second leg of the Copa del Rey semi-final to seal their place in the final for the third time in three seasons.
While there were few chances of note in the opening half-hour, Barcelona received a significant setback when Frenkie de Jong was forced off in the 26th minute.
The Dutchman was replaced by youngster Fermin Lopez, who was involved in Barcelona's brightest moment of the first period.
Unai Simon came racing out of his goal to deny Fermin, before he quickly had to get back to get fingertips to Joao Cancelo's audacious long-range effort, before Alvarez cleared the ball off the line.
The visitors were handed another injury blow in the closing stages of the first period, with Pedri unable to make it to half time due to an apparent muscle issue.
Unfortunately for those in attendance at San Mames, the entertainment value failed to improve after the break, as the two sides continued to cancel one another out.
In fact, the second period produced one shot on target, highlighting the lack of quality on display in the final third.
As the encounter ticked into stoppage time, Dani Garcia was presented with an opportunity to snatch all three points from a corner, but the midfielder could only send his header harmlessly wide of the target.
Ernesto Valverde's charges at least managed to extend their unbeaten home run to 16 matches in all competitions, while Barcelona are still yet to lose a La Liga away game this season.