Meanwhile, Sergio Romo, who's been on the DL since April 15 with a similar injury, was leaving the team for several days after the death of his grandfather, Bochy said.
Romo was supposed to throw a bullpen session before Friday night's 4-3 victory over the D-backs at Chase Field, but that didn't happen.
Kontos worked four rehab-assignment relief appearances for the River Cats, including two-thirds of inning Thursday night and an inning Friday, both against Reno, the D-backs' top farm club. He allowed a run on five hits in 3 2/3 innings, walking none and striking out two in his four appearances.
Kontos had a big smile on his face upon his arrival in the visiting clubhouse and proclaimed his arm fit.
"He came out of it feeling good," Bochy said. "There's a good chance we won't activate him until San Diego. Could we do it tomorrow, Sunday? Yeah."
About Romo, Bochy added: "He'll be here today to get his work in and then he's taking off."
Barry M. Bloom is a national reporter for MLB.com and writes an MLBlog, Boomskie on Baseball. Follow @boomskie on Twitter. This story was not subject to the approval of Major League Baseball or its clubs.