Something very similar also happened to the Panchatantra (itself a secularized compendium of many Jataka tales). Arabic versions of the Panchatantra tales were set down in a manuscript called Kalilah and Dimnah (corruptions of Karataka and Damanaka, two recurring jackals in the Panchatantra stories). In the seventh and eight centuries A.D., Jewish merchants translated “Kalilah and Dimnah” into Greek and other European languages. The stories floated around in the collective Western consciousness until Planudes in the 14th century A.D. set them down as Aesop's fables (no actual manuscript by Aesop has survived) .
So widespread was the influence of these folktales from the Jataka andPanchatantra, that Joseph Jacobs (1854-1916), the famous folktale scholar and writer wrote:
... as to the source of the tales that are common to all European children ... increasing evidence seems to show that this common nucleus is derived from India and India alone …. So far as Europe has a common store of fairy tales, it owes this to India. 
As Jacobs is careful to qualify, he is not claiming that every European fairy tale is from India; that’s plainly ridiculous. But it seems the common stories, the stories that every child seems to know in Europe, can be traced back to tales that originally appeared in the Jataka or the Panchatantra.
What was it about these tales that made them so popular? I think it is the humor. It is of a special sort. Consider. The Jataka tales have a bull named “Delight”, high-class crocodiles, monsters with sticky hair, strong-minded snakes, talkative tortoises and phony holy men. In one of the tales, Buddha works as a security guard, and in another, a queen gets laughed at by fish. Yes, fish. The humor in these tales is often droll, characteristic of British comedy. Wait. Characteristic of British comedy? Joseph Jacobs seems to think it is characteristic of the Indian storyteller:
Can we go further and say that India is the source of all the incidents that are held in commonby European children? I think we may answer “Yes” as regards droll incidents, the travels of many of which we can trace, and we have the curious result that European children owe their earliest laughter to Hindu wags. 
Perhaps that’s a bit much. Still, magic realism — whimsy’s pretty new dress — is still very popular among South-East Asian writers.
At any rate, it suffices to note that around the same time the Kalilah and Dimnah was being translated by the Jews, there lived in the court of al-Walid ibn Abdul Malek, the Caliph of Baghdad, a Christian monk called John of Damascus. St. John was born around 676 A.D. and died sometime between 754 A.D and 757 A.D. He wrote a series of works defending the then still-young Christian faith. The Arabs, who then ruled most of the world, were very secure about Islam (seeing it as extension of Christianity); the Caliph gave St. John a free hand. One of the good father’s books was a religious romance — the first Western one — called Barlaam and Joasaph.