What you seem to be calling a "prophecy" is not much more than an old song of Lake-town. While it does talk of the return of Thror to his realm under the Mountain, it is not necessarily prophesying a return nor is it held as such. It was, however, remembered by the people of Lake-town and believed much more so than The Master.
Some began to sing snatches of old songs concerning the return of the King under the Mountain; that it was Thror’s grandson not Thror himself that had come back did not bother them at all.
The Hobbit, Chapter X: A Warm Welcome
The entirety of the song is as follows:
“The King beneath the mountains,
The King of carven stone,
The lord of silver fountains
Shall come into his own!
His crown shall be upholden,
His harp shall be restrung,
His halls shall echo golden
To songs of yore re-sung.
The woods shall wave on mountains
And grass beneath the sun;
His wealth shall flow in fountains
And the rivers golden run.
The streams shall run in gladness,
The lakes shall shine and burn,
All sorrow fail and sadness
At the Mountain-king’s return!”
The text does however go on to suggests some newer songs had been written about the death of the dragon and the return of wealth:
Some of the songs were old ones; but some of them were quite new and spoke confidently of the sudden death of the dragon and of cargoes of rich presents coming down the river to Lake-town.
These newer song were inspired by The Master and were not highly thought of by the Dwarves. They had, of course, brought Bilbo along as a burglar for some reason.
These were inspired largely by the Master and they did not particularly please the dwarves...
Did Sauron know?
One can only guess how much Sauron knew about such songs. He was located nearby at the time at Dol-Guldur, as such it's possible spies and other evil creatures had overheard the songs/talk of the people of Lake-town. The death of Smaug was however very unlikely, King Thranduil is commended by the narrator for thinking the slaying of the dragon to be folly. As such the songs wouldn't have affected Sauron's "recruitment" of Smaug; if he had indeed intended to, at some point, recruit Smaug.
“But I expect they will all come to a bad end, and serve them right!” He (Thranduil) at any rate did not believe in dwarves fighting and killing dragons like Smaug, and he strongly suspected attempted burglary or something like it — which shows he was a wise elf and wiser than the men of the town.