I suppose it is possible that a piece of glass can be imbedded far enough into the sclera that blinking would not disturb it or cause you to feel it. However, if you suspect a piece of glass, or any foreign object for that matter, I would suggest seeing an ophthalmologist as soon as you can, even if you are not experiencing discomfort. Foreign objects in the eye should be removed as soon as possible, as they can result in eye infection otherwise.
Yes you can have glass in your eye without pain, depends on the size and position it's located. I have taken some amazingly large pieces of windshield glass out of people's eye's that they could not feel. See an ophthalmologist.
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