"Eastern Churches like Church of the East and Assembly of Jerusalem and Aramaic Peshitta Primacists believe that New Testament is written in Aramaic."
Here are some quotes of famous people and scholars about the importance of Aramaic and Aramaic Peshitta (Aramaic NT).
"Christ, after all spoke in the language of His contemporaries. He offered the first sacrifice of the Eucharist in Aramaic, a language understood by all the people who heard Him. The Apostles and Disciples did the same and never in a language other than that of the gathered faithful." - Latin Patriarch Maximus, Vatican.
With reference to the originality of the Peshitta text the Church "received the scriptures from the hands of the blessed Apostles themselves in the Aramaic original, the language spoken by our Lord Jesus Christ Himself...which has come down from the Biblical times without any change or revision." - Mar Eshai Shimun, California, Patriarch of the Assyrian Church of the East, April 5, 1957.
"I have no reason to doubt that the Peshitta is superior to that of the Greek. It was handed down to us by the Apostles through the scribes and preserved to us in our very own generation. No other version written in any other language can claim such authenticity and antiquity." - Patriarch Mar Dalin I, China in the 1800's, Assembly of Jerusalem
"Undoubtedly the Peshitta, written in the Aramaic language of the East, contains the pure and untainted Word of the Messiah." - Mar Yokhanan Dalin III, Portugal in 1980, Assembly of Jerusalem
"We have in the Aramaic Peshitta the preserved word of Our Lord unchanged from the time of the Apostles." - HH Patriarch Mar Michai, Detroit in 1989, Assembly of Jerusalem.
One topic often used as supporting evidence of Greek primacy, is that many of the important early Christians were Greek, such as Timothy and Titus. The Greek NT says that they were Greek (in Acts 16:1 and Galatians 2:3 of Greek NT), but the original Aramaic NT (Aramaic Peshitta) tells us that they were actually Arameans (in Acts 16:1, Galatians 2:3 of Aramaic Peshita). Arameans are Aramaic speaking Gentiles.
In Hebrew, "Ben" means Son. But in Aramaic, "Bar" means Son.
By the time of Jesus, Aramaic was the language of Israel. Not Hebrew. Just look at the names in English NT Bible- "Bar"tholomew, "Bar"abbas, Simon "Bar" Jonah, "Bar" Jesus, "Bar"nabbas, "Bar"sabbas, "Bar"timaeus, etc. A common female name during the time of Jesus was Martha which in Aramaic means "Mistress or Lady."
In Josephus' Jewish Wars, we see that Simon Bar Giora was one of the three leaders who fought against Romans in Jewish Wars. "Bar Giora" means "son of a proselyte" in Aramaic.
In Antiquities of Jews (XX, XI), Josephus points out that Jewish Nation didn't speak Greek language and they didn't encourage the learning of Greek language.
According to Josephus, the Romans had to have him translate the call to the Jews to surrender into "their own language" (Jewish Wars 5:9:2).
In Jewish Wars (Book 1, Preface, Paragraph 1), we also read that Josephus wrote "Jewish Wars" formerly in the language of his country before he translated it into Greek tongue for the sake such as live under the government of Romans.
In Our English Bible, We see Jesus speaking Aramaic. For Example, We see Jesus saying "Talitha Cumi." Talitha is Aramaic. If it was Hebrew, then "Yaldah" would have been used instead of Talitha.
People might ask why does it say Hebrew instead of Aramaic in New Testament? When Jews say "Hebrew", they are referring to their "Hebrew tongue." Just like, we call Deutsch "German", because it is the language of German People. Jews are called Hebrews, because they are the descendants of Abraham who is called a Hebrew (Genesis 14:13, Philippians 3). Even today, many Iraqi jews call their aramaic "hebrew" (Hebrew is called "Ibraith" in Aramaic), because it is the language of Hebrews.
But you may ask how can Apostle Paul write Aramaic to People in Greek regions like Ephesus, Corinth, Thessalonica, etc? From Acts 16 to 18, we read that Apostle Paul preached at the Jewish synagogues in Greek regions. So we see that there were Jewish communities living in Greek regions during that period. We also read that Jews lived in Rome until Claudius Caesar (41-54 AD) ordered the Jews to leave Rome in Acts 18.
For more informations on Peshitta Primacy, here is the link.