Let us first consider the most important external, i.e. non-Islamic, influences.

I. CHRISTIANITY
It is obvious that the ascetic and quietistic tendencies to which I have referred were in harmony with Christian theory and drew nourishment therefrom. Many Gospel texts and apocryphal sayings of Jesus are cited in the oldest Sufi biographies, and the Christian anchorite (rahib) often appears in the rôle of a teacher giving instruction and advice to wandering Moslem ascetics. We have seen that the woollen dress, from which the name 'Sufi' is derived, is of Christian origin: vows of silence, litanies (dhikr), and other ascetic practices may be traced to the same source. As regards the doctrine of divine love, the following extracts speak for themselves:
"Jesus passed by three men. Their bodies were lean and their faces pale. He asked them, saying, 'What hath brought you to this plight?' They answered, 'Fear of the Fire.' Jesus said, 'Ye fear a thing created, and it behoves God that He should save those who fear.' Then he left them and passed by three others, whose faces were paler and their bodies leaner, and asked them, saying, 'What hath brought you to this plight?' They answered, 'Longing for Paradise.' He said, 'Ye desire a thing created, and it behoves God that He should give you that which ye hope for.' Then he went on and passed by three others of exceeding paleness and leanness, so that their faces were as mirrors of light, and he said, 'What hath brought you to this?' They answered, 'Our love of God.' Jesus said, 'Ye are the nearest to Him, ye are the nearest to Him.'"
The Syrian mystic, Ahmad ibn al-Hawari, once asked a Christian hermit:
"'What is the strongest command that ye find in your Scriptures?' The hermit replied: 'We find none stronger than this: "Love thy Creator with all thy power and might."'"
Another hermit was asked by some Moslem ascetics:
"'When is a man most persevering in devotion?' 'When love takes possession of his heart,' was the reply; 'for then he hath no joy or pleasure but in continual devotion.'"
The influence of Christianity through its hermits, monks, and heretical sects (e.g. the Messalians or Euchitæ) was twofold: ascetic and mystical. Oriental Christian mysticism, however, contained a Pagan element: it had long ago absorbed the ideas and adopted the language of Plotinus and the Neo-platonic school.

II. NEOPLATONISM
Aristotle, not Plato, is the dominant figure in Moslem philosophy, and few Mohammedans are familiar with the name of Plotinus, who was more commonly called 'the Greek Master' (al-Sheykh al-Yaunani). But since the Arabs gained their first knowledge of Aristotle from his Neoplatonist commentators, the system with which they became imbued was that of Porphyry and Proclus. Thus the so-called Theology of Aristotle, of which an Arabic version appeared in the ninth century, is actually a manual of Neoplatonism.
Another work of this school deserves particular notice: I mean the writings falsely attributed to Dionysius the Areopagite, the convert of St. Paul. The pseudo-Dionysius--he may have been a Syrian monk--names as his teacher a certain Hierotheus, whom Frothingham has identified with Stephen Bar Sudaili, a prominent Syrian gnostic and a contemporary of Jacob of Saruj (451-521 A.D.). Dionysius quotes some fragments of erotic hymns by this Stephen, and a complete work, the Book of Hierotheus on the Hidden Mysteries of the Divinity, has come down to us in a unique manuscript which is now in the British Museum. The Dionysian writings, turned into Latin by John Scotus Erigena, founded medieval Christian mysticism in Western Europe. Their influence in the East was hardly less vital. They were translated from Greek into Syriac almost immediately on their appearance, and their doctrine was vigorously propagated by commentaries in the same tongue. "About 850 A.D. Dionysius was known from the Tigris to the Atlantic."
Besides literary tradition there were other channels by which the doctrines of emanation, illumination, gnosis, and ecstasy were transmitted, but enough has been said to convince the reader that Greek mystical ideas were in the air and easily accessible to the Moslem inhabitants of Western Asia and Egypt, where the Sufi theosophy first took shape. One of those who bore the chief part in its development, Dhu ’l-Nun the Egyptian, is described as a philosopher and alchemist--in other words, a student of Hellenistic science. When it is added that much of his speculation agrees with what we find, for example, in the writings of Dionysius, we are drawn irresistibly to the conclusion (which, as I have pointed out, is highly probable on general grounds) that Neoplatonism poured into Islam a large tincture of the same mystical element in which Christianity was already steeped.

from:
THE MYSTICS OF ISLAM
by Reynold A. Nicholson
Routledge, Kegan Paul, London