Why are the manuscripts of the New Testament in Greek when Jesus spoke Aramaic?

We don't know why God chose the Greek language for inspiration, but we should not lay restrictions on Him. However, it was the universal language of that time right up until1000 AD.

When we read the Qur'an, we find verses in Arabic which are said to be the words of Jesus (Surah 3:49-54; 5:114-118; 61:14 etc.). Indeed Muslims believe that they are words of Jesus. Now, if Jesus' words can be recorded as the word of God in Arabic, a translation, then surely his words can also be recorded in the Greek language of the Christian Scriptures.

One may also find words spoken by various prophets in the Qur'an. Most of these prophets spoke Hebrew but their words are also recorded in Arabic in the Qur'an. So one can see that Muslims don't have these people's original words either. In spite of this, if such a situation does not effect the integrity of the Qur'an, then nor should it be treated as evidence to question the integrity of the Bible.