Yes, warts can bleed, although it's not typically a frequent occurrence.
When warts are located in areas that experience friction, such as those rubbed against clothing or frequently bumped, they can become irritated. This irritation can cause the skin around the wart to become painful and, in some cases, lead to bleeding. Because warts have their own blood supply, damage to the wart itself can also result in bleeding. While most warts eventually resolve on their own within a year or two, persistent or recurring warts are more likely to be subjected to trauma and subsequent bleeding. If bleeding is a recurring issue, consult a doctor to determine the best course of treatment.