There is unrest at Rugby Park owing to the departure of manager Kenny Shiels, with some quarters of support blaming Johnston.
In a statement issued jointly under the name of 'Ayrshire Businesses', a meeting was requested with Johnston to discuss the best course of action for the club.
It read: "Tonight a group of like-minded business people from Kilmarnock met to discuss their concerns about the current state of affairs at Kilmarnock Football Club.
"The local businesses represented were Braehead Foods, Brownings the Bakers, The Coffee Press, Costley & Costley Hoteliers, Fortress Security & Alarms, Hamilton Tarmac, The Klin Group, Microtech Support, Modway and the QTS Group.
"The group has unanimously agreed that the only way they can continue to fully support the club is if there is a change in regime.
"A spokesperson has been appointed to request a meeting with Michael Johnston to clarify his position and discuss the way forward in the best interest of Kilmarnock Football Club."
Protesting supporters, unhappy at Shiels' sacking last week, have hoped the Klin Group will head a takeover of the club from Johnston.
- Kilmarnock Football Club
- Kenny Shiels