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I think it was most probably Aramaic. This is the common language of Judea in the first century AD, most likely a Galilean dialect distinguishable from that of Jerusalem. This is generally agreed upon by historians. The villages of Nazareth and Capernaum in Galilee, where Jesus spent most of his time, were Aramaic-speaking communities. Most probably Jesus' disciples spoke Aramaic, too. Remember those who accused Peter of being from Galilee because of his accent? Matthew 26:73 says in the New Living Translation, "A little later some of the other bystanders came over to Peter and said, 'You must be one of them; we can tell by your Galilean accent.'” Source for some of this: Wikipedia
Scholars agree that Jesus as a resident of Israel would have spoken Hebrew, Aramaic, and Greek. However, they disagree what language it was that He primarily used. It is traditionally thought that Aramaic was the language because it is believed spoken Hebrew died out due to the Babylonian Captivity and that any use was limited to the temple and religious leaders. It is said that Aramaic replaced Hebrew. But history and the Scriptures indicate Hebrew was still a living language in Jesus’ day. In some versions, the word ‘Aramaic’ is used for the Greek word ‘hebraisti.’ That translation is not valid because ‘Hebrew’ is the meaning. There is a Greek word for Aramaic, which is ‘Syristi,’ but it is not found in the New Testament. Some Aramaic words are the same as Hebrew words, but Hebrew is not Aramaic or close to it as II Kings 18:26 attests. ‘Gabbatha,’ John 19:13, ‘Golgotha,’ John 19:17, ‘Bethesda,’ John 5:2, are all specifically said to be ‘hebraisti’ words. The inscription over the cross was in Hebrew (‘hebraisti’), Greek and Latin, John 19:20. Later, Luke explained to readers that Jesus spoke to Saul in Hebrew, Acts 26:14, and that Paul spoke in Hebrew in Acts 21:40, 22:2. In these cases, it clarified to Greek readers that it was his mother tongue. The same word ‘Hebrew’ is in II Corinthians 11:22, Philippians 3:5, and Revelation 9:11, 16:16. Plainly, ‘Hebraisti’ is not Aramaic but Hebrew. In Mark 5:41, Jesus said to the girl, “talitha, cumi.” The words are Aramaic, but it is also Hebrew, transliterated into Greek: ‘Little lamb, rise up.’ When Jesus spoke to the deaf man with the speech impediment, “ephphatha,” Mark 7:34, it seems to be Aramaic but it is not. It is a Hebrew word transliterated into Greek: ‘be opened.’ Many other words usually attributed to Aramaic are actually Hebrew words: ‘mammon,’ Matthew 6:24, ‘corban,’ Mark 7:11, ‘raca,’ Matthew 5:22, ‘abba,’ Mark 14:36, ‘hosanna,’ Matthew 21:9, and ‘rabboni,’ John 20:16. “Eloi, Eloi, lama sabachthani,” Mark 15:34, is the Greek transliteration of the Hebrew. Doug Hamp explains this well in “Discovering the Language of Jesus: Hebrew or Aramaic?” Hebrew had to be the language Jesus used when He interacted with the teachers in the temple, Luke 2:46-47, and in reading the Hebrew Scriptures, Luke 4:16. There is no reason to think He did not use it in all of His ministry, as it was the language of Israel at that time. As the New Testament was written for a wider audience, the writers penned their work in Greek, the universal language of the Roman Empire. They would use the Greek Septuagint for quotes from the Hebrew Scriptures Jesus used. It is possible some of the gospels, and perhaps Hebrews, were originally in Hebrew, later translated into Greek. Papias (c130 AD) wrote, “So then Matthew wrote the oracles in the Hebrew language, and everyone interpreted them as he was able.” Of the several languages spoken in Israel, Hebrew was the main language Jesus spoke.
Though the above answers do a much better job of directly answering your question assuming you were asking what language did Jesus speak in the recorded events listed in the New Testament, I want to add that Jesus spoke (and still does) all languages. Much as the story in Acts chapter 2 illustrate where everyone heard the messages in their own language (as translated by the Holy Spirit), Jesus throughout history has answered prayers in the language of people from all nations and continues to today. Everyone can speak to Him without having to learn a new language. And hear His answers to those prayers when they take the time to listen!
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