[Legate of Amon Lanc]

That's quite interesting thought. But could he possibly have told Sauron? I wonder if Saruman, being the part of the White Council and all, had known about the Elven Rings...
We know that the Rings were his domain, he studied all he could about them, as Gandalf himself says in the Council of Elrond:

The lore of the Elven-rings, great and small, is his province. He has long studied it, seeking the lost secrets of their making...

And since it was well known that there were some "Three rings to the elven kings under the sky", it is safe to assume that Saruman might be interested also in their whereabouts. We know that Saruman discovered that Gandalf had Narya, as is written in the Unfinished Tales:

And the Grey Messenger took the Ring, and kept it ever secret; yet the White Messenger (who was skilled to uncover all secrets) after a time became aware of this gift, and begrudged it, and it was the beginning of the hidden ill-will that he bore to the Grey, which afterwards became manifest.

However, we don't know what Saruman knew about the other two Rings. I am quite convinced that Elrond and Galadriel never told him about the Rings they had. The Rings were secret even between the elves of Rivendell and Lórien, although the inhabitants might have suspected something like that, it was never explicitely told that the Rings really are there. Galadriel revealed her Ring to Frodo as if it had been a great mystery:

'Yes,' she said, divining his thought, `it is not permitted to speak of it, and Elrond could not do so. But it cannot be hidden from the Ring-bearer, and one who has seen the Eye. Verily it is in the land of Lórien upon the finger of Galadriel that one of the Three remains. This is Nenya, the Ring of Adamant, and I am its keeper.' - The Mirror of Galadriel

Galadriel, I think, speaks to us very clearly in this point. The rings are tabu. And as we also know, she did not like Saruman, so even if she wanted to tell someone about the Ring, it would certainly not be him.
Continuing this speech, Galadriel also speaks about Sauron, that

`He suspects, but he does not know – not yet.'

We might assume the same about Saruman. He could have suspected, but he wouldn't know for sure. On the other hand, we are told that Saruman was "skilled to uncover all secrets" and that he revealed Gandalf's possession of Narya, hence, it is not impossible that similar ways he discovered the truth about Elrond and/or Galadriel's Rings.

But even if Saruman had known about the Rings, I am not sure if he would have revealed that knowledge to Sauron. Personally I think he would not - he didn't tell Sauron about the Shire, for example, and many other things. Saruman played with his own cards, and he wanted to claim the One Ring for his own, he had no need to tell Sauron - even if he was in situation where he needed to "buy" Sauron's trust by providing information, I'm sure he could have used some less important information to satisfy the questioner. And if he would have been explicitely asked, he could have repeated the words of Galadriel: "I have my own suspicions, but I am not sure, not yet. Perhaps later, yes m'lord, surely I'll look into this."

This wouldn't be of much use to Sauron, because from what we are told in the UT (The History of Galadriel and Celeborn, "Concerning Galadriel and Celeborn"), Sauron had the same suspicions already from the Second Age:

Concerning the Three Rings Sauron could learn nothing from Celebrimbor; and he had him put to death. But he guessed the truth, that the Three had been committed to Elvish guardians: and that must mean to Galadriel and Gil-galad.

So the only thing Sauron wouldn't know would be if Galadriel received two Rings and Gil-galad one, or vice versa, or if someone else got the third, and where the Rings have gone after the death of Gil-Galad.

And the last thing. Even if Sauron had learned the whereabouts of the Rings, it wouldn't be of much use to him, it would only help him in marking the most important targets of his armed assault. Only if he (or Saruman) gained the possession of the One Ring, they will be able to uncover and destroy all that has been done with the Elven Rings.