According to VCA Animal Hospitals, it is possible to contract rabies or another illness through cat scratches. The only way to determine if an aggressive cat has rabies is to quarantine the animal or take a brain tissue sample.
Johns Hopkins Medicine notes that, although rare, animals do transmit the rabies virus through deep puncture wounds like scratches. For this to occur, the rabid animal must groom its paw in order to get infected saliva onto the claws. When the animal scratches a human soon after grooming, the rabies virus may pass on from the animal's saliva into the bloodstream of the human. It is important that any person who thinks that they might be infected with rabies to seek immediate medical attention for diagnosis and treatment.