There seems to be two main paths of supposed influence:
1) the typical Hellenistic cosmological concept, involving the Logos (the Word), which some would say has influenced the person of Jesus Christ in Christianity and Islam. There is almost no doubt that Jesus being titled the word or a word of God in the Quran is due to Christian influence, especially when we consider church history. Then we need to conclude whether John intended the use of the Logos as contextualism, or whether, as some say, that John referred to the word as God's wisdom (Proverbs 8), or as some say that John is using Philo (who fused Judaism and Hellenism) to target Jews and Gentiles alike.
There is no indication that John intended the use of Logos as more than a means of contextualisation, sadly, later Christian theologians and philosophers such as Justin Martyr and later Origin jumped into a fusion of Christianity and Greek philosophy, thus the Logos in a Christian context was thrown into a sea of speculation.
2) ancient religions and mystery religions. Some bring up the idea that Jesus was born by a virgin on the 25 December much alike Mithra and other religious figures, that Jesus was attended by shepherds much like Mithra, that he was a miracle maker, that he had twelve disciples much like Mithra, that he died in the same manner and for the same purpose much like Mithra, Osiris and other mythological figures, etc, that he was taken to heaven much like etc, etc...
If this is true, that 3-4 Christians did borrow such elemens from mystery religions, it would certainly bring Christianity into serious difficulties and Islam as well, that is of course, unless the Muslim wants to pick and choose from atheist scholars what belongs to Christianity only and which leaves out Islam.
However, what do Muslims and Christians on this forum have to say about the matter?
Be blessed
Kai