The full-back, who starred for the Dons in the 1-0 Betfred Cup semi-final victory over Steven Gerrard’s men, was filmed by one of his friends bragging about the result.
Logan was heard to say: “He bought my dinner, we f****d his team yesterday, the most I can do is let him come round and watch the football at my house. F*** the Rangers.”
According to the Daily Record, Aberdeen have started an internal investigation and may look to punish the player for his actions.
It is also understood that the SFA will be reviewing the evidence and may decide to impose sanctions on the 30-year-old.
Logan missed the first three games of this season after being sent off in Aberdeen’s 1-0 victory away to Celtic in the final game of last season. The Englishman goaded the Parkhead crowd and clashed with several opponents as he left the field.