I'm not an expert on this one, but most sources do suggest that -- whatever there actual origin --the Linking Rings were introduced into Europe early in the nineteenth century by a troupe of Chinese magicians. Several accounts credit what is believed to be the first company of Chinese "jugglers" that appeared in London's West End in 1830 with their introduction. We know that the effect was quickly adopted by Western conjurers. The English magician M. Jacobs (18??-1870)performed the Linking Rings as early as 1834. Two years later, the great French conjurer Philippe (Jacques Andre Noel Talon 1802-1878) added the effect to his repetoire, claiming to have been taught the trick by these Chinese magicians whom he met in Dublin. Whether this is true or not (see Pete's posting above), the fact remains that the Linking Rings quickly became a high point of Philippe's act.
As for whether the effect actually has a Chinese link, Reverend Doolittle's 1867 work The Social Life of the Chinese recounts his years as a missionary in China and notes the Linking Rings as a common form of popular entertainment at the time. Of course, I'd accept Pete's view that the effect may well be early Egyptian in origin. After all, he was there :D
I hope this helps.