SFA chief Ian Maxwell has written to all 10 Championship clubs to ask if the season should be suspended.
All football below the top two divisions of the men's game has been suspended for at least three weeks to act as a fire break amid the ongoing coronavirus pandemic.
Championship clubs are now required to test once a week and an outbreak of Covid-19 has been detected at Raith Rovers.
That saw their fixture against Inverness on Saturday postponed and according to the BBC the SFA are now considering putting the second flight on ice too.
Chief executive Maxwell is said to have sent an email to all 10 clubs asking if they want to stop the season, with a Monday deadline for a decision to be made.
The correspondence states that one club wants all football to stop though it's not clear to which team he is referring.
Maxwell wrote: "One SPFL championship club has publicly taken the position that all football should be suspended.
"The Scottish FA would like to understand the position of your club... please confirm if you are content to continue under the current exemption or whether your club considers that the Championship should be suspended."
