The Greek word for offense is σκάνδαλον which is the origin of the English word scandal. Scandals are things that we find offensive, often for good reason, sometimes less so. We can, for example, be offended at something good.
The word σκάνδαλον appears in the New Testament fourteen times, almost half of those in the mouth of Jesus. He notes that "it is unavoidable that scandals not come but woe to those through whom they come" (Luke 17:1), or as Matthew puts it: "Woe to the world because of scandals; it is necessary that scandals come but woe to the man through whom the scandal comes" (Matthew 18:7). At one point, Jesus tells Peter: "Go behind me, Satan, for you are a scandal to me because you consider not the things of God but the things of man" (Matthew 16:23).
In relating the parable of the wheat and the tares, Jesus notes that both the wheat and the tares grow together, and they are hard to distinguish, but in the end, "the Son of Man will send his messengers and they will gather out of the kingdom all the scandals and those who work iniquity" (Matthew 13:41).
So for Jesus, seeking after the things of the world and working iniquity are scandals that merit being cast out of the Kingdom.
The verbal form---meaning to scandalize, cause a scandal, to cause offense, and in the passive often shades into meaning to apostatize---appears more often, thirty times in the New Testament, twenty-six of them in the mouth of Jesus.
Jesus says that those who cause a scandal should be cut off and cast off (Matthew 5:29-30; 18:8-9); he says this more than once, one of those in the context of immorality but not the other. In giving the parable of the sower, Jesus notes that some receive the word gladly but they are temporary and have no roots in the gospel; the minute that affliction or persecution come, they are scandalized (Matthew 13:21). Jesus said the words "A prophet is not without honor except in his own country" to his brothers who were scandalized by him (Matthew 13:57). Jesus's disciples noted that the Pharisees were scandalized by the things that he said (Matthew 15:12). He also warned those who scandalize those who believe that they will be held accountable for it and that they would actually be better off dead (Matthew 18:6). Jesus tells his disciples that in the last days "then they will betray you to affliction and kill you and you will be hated by all nations because of my name, and then many will be scandalized and betray each other and hate each other and many false prophets will arise and deceive many" (Matthew 24:9-11). Therefore, he said, "Blessed is he who is not scandalized by me" (Matthew 11:6).
So Jesus says quite a bit about scandals, and especially warns about those who cause scandals among believers because they seek to be accepted of the world, or are immoral, or betray and hate the believers. In the end, Jesus will get rid of the scandals and those who cause them. In the end, tares will not be able to pretend they are wheat.