The veteran first baseman, signed in early March and assigned to Double-A Akron, is headed for Triple-A Columbus.
Acta said not to expect Johnson with the big league club anytime soon.
"I'm not going to speculate on it," Acta said. "At this point, he's not even on our 40-man roster."
Johnson, who has had four surgical procedures on his wrist in the last year, played just 24 games last season.
"We want to take a look at him," Acta said. He's going to be playing some games in Columbus. He's got a lot of at-bats to go before we make a decision."