I would think as long as it's clear it should be his eye's way of adding lubrication to his eye. If it turns green or yellow he should get it checked.
Next time you are at the vet you can have his eyes checked for tear production, he may be producing thicker mucus for his eyes due to dry eye, which is common in bulldog breeds.
By the way,and Cheddar is adorable, that avatar is too cute!