When Jesus entered Capernaum he healed the slave of a centurion. Did the centurion come personally to request Jesus for this? At first glance, there appears to be a contradiction between Matthew 8:5, which seems to imply that the centurion came to Jesus, and Luke 7:3 and 7:6, which say he did not.
Now when Jesus had entered Capernaum, a centurion came to Him, pleading with Him, saying, "Lord, my servant is lying at home paralyzed, dreadfully tormented." And Jesus said to him, "I will come and heal him." The centurion answered and said, “Lord, I am not worthy that You should come under my roof. But only speak a word, and my servant will be healed. (Matthew 8:5–8)
So when he [the centurion] heard about Jesus, he sent elders of the Jews to Him, pleading with Him to come and heal his servant. And when they came to Jesus, they begged Him earnestly, saying that the one for whom He should do this was deserving, “for he loves our nation, and has built us a synagogue.” Then Jesus went with them. And when He was already not far from the house, the centurion sent friends to Him, saying to Him, "Lord, do not trouble Yourself, for I am not worthy that You should enter under my roof. (Luke 7:3–6)
In Luke 7:6–8 we see the deployment of the centurion’s friends, who spoke the thoughts of his own heart. Matthew mentions the case, but far more succinctly. Matthew frequently abbreviates his retellings of events. [See Matthew 8:14–15 (cf. Mark 1:29–31); Matthew 9:1–8 (cf. Mark 2:1–12); Matthew 9:18–26 (cf. Mark 5:21–43); and Matthew 11:2–6 (cf. Luke 7:18–23)]. So at first blush, we might conclude from Matthew’s account that the centurion came himself; whereas it is clear there was the intervention of both Jewish elders and friends from Luke’s account. Matthew omits the elders’ praise of the centurion in his narrative, but Matthew does include the centurion’s elders and friends, only he does it indirectly and in a unique way. The clue to it is that old and well-attested maxim of law that what one does by another one does by oneself (often mentioned in Scripture: 2 Samuel 3:18, 14:19; 2 Kings 14:27; Esther 8:8; Ezra 1:7–8). So although the narrative in Matthew does not directly mention the centurion’s friends and elders, he does so indirectly by summing it all up, simply speaking of the friends’/elders’ words as nothing but a projection of the centurion’s pleas. Matthew skips the intermediaries and in verse 10 records Jesus piercing to the thoughts and intents of the centurion’s heart.
When looking at the two texts in total (Matthew 8:5–13 and Luke 7:1–10), it seems quite likely that when Jesus came near to the centurion's house (Luke 7:6), He was probably within shouting distance. Consequently, when Jesus was talking to the centurion's friends, He either spoke loudly enough that the centurion could hear (and was thus addressing him directly) or he spoke to the friends as people who were direct mouthpieces for the centurion. Luke 7:9 states that while talking to the centurion's friends Jesus marveled at him (i.e., the centurion), so it is quite possible that Jesus was talking to the friends of the centurion, but looking directly at and addressing the thoughts and intentions of the centurion. The centurion may have been mindful that Jesus would be considered ceremonially impure if He came into a Gentile’s house (as mentioned in a different context in Acts 10:28) and therefore stood outside the house, so that if Jesus persisted in coming to the centurion, He would not be defiled. Remember that on another later occasion (Matthew 16:23) Jesus addressed someone directly who was the controlling force behind another, even though they were not present (“get behind me Satan” to Peter).
In Luke 7:4–6 Jesus listened to and then went with the elders of the Jews, toward the centurion’s house, after hearing their request and their reasons for it without any reluctance; He at once complied, did not hesitate, or raise any objection about it, but went with them freely. Matthew completely omits this detail and discourse and instead focuses on the centurion’s thoughts and motives, not what his friends and benefactors thought of the centurion. Remember that Luke was writing to a Gentile audience. Luke wanted Theophilus to be aware that most Jewish people respected a God-fearing Roman soldier. Matthew writing to a Jewish audience wanted them to know that a Roman centurion was sincerely respectful of Jesus. So when viewed with these different perspectives and audiences in mind, it becomes clear that we should expect different emphases from the writers, but the account itself is non-contradictory.