Lionel Messi had Barca’s best chance as his lob from outside the box hit the top of the crossbar.
The hosts were the better team and Barca keeper Marc-Andre ter Stegen pulled off excellent saves to deny Markel Susaeta and Inaki Williams.
Athletic ended with 10 men when Oscar de Marcos was shown a second yellow card in injury time for handball.
Champions Barcelona, who are unbeaten in 11 league games, missed the chance to go eight points above second-placed Real Madrid, who beat Atletico 3-1 in the capital derby on Saturday.
It was the first time they had failed to score in a La Liga match in 38 games, since a goalless home draw with Getafe on 11 February 2018.