Boss Wilson and head coach Greenwood had been at Priory Lane for 27 years between them.
First team coach Steve Eke has also left the club.
The decision comes after a disappointing run of results with Borough currently 16th in Blue Square Bet south.
Wilson said: “Obviously I’m disappointed it has come to this. I still feel the targets the club had set us as a management team were more than achievable."
Club captain Ben Austin has been placed in caretaker charge until the club make an appointment.
For more reaction see The Argus tomorrow.
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