Bilbao took an early lead through Ander Capa at the Ramon Sanchez Pizjuan and while an own-goal from Unai Nunez brought Sevilla level on the hour, the home side were unable to find a winner.
It means Sevilla trail Real Madrid, who they play at the Santiago Bernabeu on January 18, by two points and Barcelona by four.
Those gaps will grow on Saturday if Madrid win away at Getafe and Barcelona overcome local rivals Espanyol.
With Atletico Madrid struggling for goals and Valencia still recovering after removing their coach, Sevilla have looked like the team most likely to challenge Spain’s established duo.
But after an encouraging run of form in October and November, Julen Lopetegui’s side have managed only one win from their last four league matches.
Qualification for the Champions League may soon be the best they can hope for, which would be no mean feat after they signed 13 players and sold 15 last summer, as well as hiring a new coach.
Athletic Bilbao are in the hunt too. They sit seventh, three points behind fourth-placed Atletico Madrid, who play at home to Levante on Saturday.
Gaizka Garitano’s side took the lead in the 15th minute after an excellent control and finish from Capa, with Kenan Kodro finding the net again shortly after only for the goal to be ruled out for offside.
Sevilla turned the screw in the second half.
Jesus Navas, making his 500th appearance for Sevilla, almost set up an equaliser only for Luuk de Jong’s finish from six yards to come back off the post.
The home did level with half an hour left. Lucas Ocampos drove to the line and his cross was turned in by Nunez at the near post.