Barcelona have handed Real Madrid a boost with their goalless draw against Athletic Club, as their lead at the top of LaLiga Santander is now just six points.
The hosts were the better team and Barca keeper Marc-Andre ter Stegen pulled off excellent saves to deny Markel Susaeta and Inaki Williams.
Barca missed the chance to move eight points clear of second-placed Real Madrid, but did manage to restore their six-point gap at the top.
Ernesto Valverde's men returned in much more positive mood and Messi curled a 53rd-minute free-kick just over, although the Argentina worldwide was growing increasingly irritated by Bilbao's physical approach to marking him.
After Real Valladolid at home next weekend, Valverde's side travel to Lyon in the Champions League and Sevilla in the league.
Ousmane Dembele made his recovery from an ankle problem as a 75th-minute substitute, taking the place of Coutinho, whose problems in a Barcelona shirt continued.
Barca enjoyed more control in the second half after homegrown midfielder Carles Alena replaced an erratic Arturo Vidal but exhausted as the game progressed.
First, Suarez peeled free at the back post but glanced wide from Messi's cross and then Messi hit the crossbar, the goal gaping after Iago Herrerin had rushed out to intercept Nelson Semedo.
The game was frantic and Bilbao were more than holding their own.
Suarez stabbed wide with the outside of his right foot after turning on Messi's 74th-minute pass, but Gerard Pique did just enough at the other end to prevent Williams from making meaningful contact with substitute Iker Muniain's free-kick five minutes later.
Bilbao had De Marcos sent off for a second bookable offence - deliberate handball - in the final moments, but Barcelona, who were denied a late penalty by VAR, could not find a victor as they failed to triumph for the third match in succession.
The Bilbao forward had an another glorious opportunity two minutes later, but Ter Stegen came to Barca's rescue with a superb to deny the striker in the box.