| 1. Round | 03/08/2019 | Ayr United | - | Greenock Morton FC | 4:2 | ||||
| 3. Round | 23/08/2019 | Greenock Morton FC | - | Partick Thistle | 3:2 | ||||
| 6. Round | 21/09/2019 | Greenock Morton FC | - | Dundee FC | 1:0 |  |
|||
| 7. Round | 28/09/2019 | Dundee United | - | Greenock Morton FC | 6:0 | ||||
| 8. Round | 05/10/2019 | Greenock Morton FC | - | Dunfermline Athletic | 1:1 |  |
|||
| 9. Round | 19/10/2019 | Arbroath FC | - | Greenock Morton FC | 1:0 | ||||
| 10. Round | 26/10/2019 | Greenock Morton FC | - | Inverness CT | 2:1 | ||||
| 11. Round | 29/10/2019 | Greenock Morton FC | - | Ayr United | 2:3 | ||||
| 12. Round | 01/11/2019 | Dundee FC | - | Greenock Morton FC | 2:1 | ||||
| 13. Round | 09/11/2019 | Partick Thistle | - | Greenock Morton FC | 2:1 | ||||
| 14. Round | 16/11/2019 | Greenock Morton FC | - | Arbroath FC | 1:0 | ||||
| 16. Round | 07/12/2019 | Dunfermline Athletic | - | Greenock Morton FC | 3:1 | |
|||
| 15. Round | 10/12/2019 | Greenock Morton FC | - | Dundee United | 1:2 | ||||
| 18. Round | 21/12/2019 | Alloa Athletic | - | Greenock Morton FC | 0:2 | ||||
| 19. Round | 28/12/2019 | Greenock Morton FC | - | Partick Thistle | 1:2 | ||||
| 20. Round | 04/01/2020 | Arbroath FC | - | Greenock Morton FC | 1:2 |
Scotland