BRUSSELS—The European Union could label Lebanon-based Hezbollah a terrorist organization after Bulgaria linked the Shiite organization to a deadly bomb attack there in July, an EU official said Wednesday.

The U.S., Israel and others have pushed Brussels for years to toughen its stance against Hezbollah, but France, Sweden and others have resisted the pressure. A report issued Tuesday by Bulgarian investigators reopened the debate, finding two of the alleged perpetrators in the attack, which killed five Israeli tourists, were...