The 32-year-old striker had to be replaced before half-time at San Mames after receiving treatment following a shot that hit the post.
Lionel Messi missed the match with a calf issue of his own, suffered shortly before Barca travelled to the United States for two friendly matches against Napoli last week.
It was the first time in 10 years that Barca started their league campaign without the Argentine and both players' absences were felt as Ernesto Valverde's men succumbed to a stunning late winner from Athletic veteran Aritz Aduriz.
After he was treated on the field, Suarez was ultimately unable to continue and was taken off in the 37th minute, replaced by Rafinha.
"Tests carried out on Saturday morning have confirmed Luis Suarez has an injured right calf," Barca said in a statement.
"His recovery will dictate his return to action."
Suarez's early withdrawal was the first time he has been substituted in the first half of a La Liga game since he joined Barca from Liverpool in 2014.
The Uruguayan netted 25 times in all competitions for the Catalan giants last season and will look to play another pivotal role in their ambitions to retain the title when he returns from injury.
The champions suddenly look light on forwards, with Philippe Coutinho set to join Bayern Munich on loan with a view to a permanent deal.
However, Messi is making good progress according to head coach Ernesto Valverde, although he is yet to train with the team.
The forward did not feature in any of Barca's pre-season matches, after returning late from international duty at the Copa America.
Messi and Suarez netted 76 goals between them last season, with goalscoring duties now likely to fall to Antoine Greizmann and Ousmane Dembele as Barca prepare for their next league encounter against Real Betis.
Griezmann debuted against Athletic, but Suarez's injury and Coutinho's departure could see Barca ramp up their interest in former player Neymar, who looks increasingly likely to leave Paris Saint-Germain.