The 69-year-old returned to the club shortly after Cheltenham were relegated back to the fourth tier at the end of 2023-24 and following the departure of manager Darrell Clarke.
Johnson was then involved in the appointment of head coach Michael Flynn, who guided the Robins to a 15th-place finish last season.
The club signed 11 players this summer, including bringing defender Robbie Cundy in and re-signing midfielder Ryan Broom.
Yet Johnson has come in for criticism from fans in recent weeks as the team have struggled on the pitch, with a banner calling for him to leave at their most recent home game.
Cheltenham were taken over by Mike Garlick last month, with the former Burnley chairman taking a “controlling interest” and joining the club’s board.