Yes, at most occasion, having a cold will make you bloodshot in your eye, especially when you have a large amount of sleep, so at that time, pls take care of you and your eyes. Of course, when you found that your eyes are bloodshot, you cannot say it must be caused by the cold, because there are many reasons that will make your eyes bloodshot, such as the lacking of vitamins and conjunctivitis or pink eyes. By the way, when you notice your eyes are red, it should be bloodshot, and also it should not be, because when you have cold, the function of immune system will become weaker and then your eyes are easier to get infected. If so, you eyes are the same red as the bloodshot. I want to tell you that if you get cold, you should keep warm and keep on exercise in order to strengthen the immune system, which will help you to fight with the bacteria.