That is in truth why I brought you here. For I have many eyes in my service, and I believe that you know where this precious thing now lies. Is it not so? Or why do the Nine ask for the Shire, and what is your business there?
Most of us will recognize this as what Saruman said to Gandalf at the moment of his capture in Orthanc.
Not only these words, but also the bold act of imprisoning Gandalf meant that Saruman was aware that:
1. the one Ring was found, and that
2. Gandalf withheld information on its whereabouts.
As regards the 2d point, since the Nazgûl had been inquiring about the Shire, a quick association with Gandalf was easy to make. But how did Saruman know about the 1st point? JRRT left that part to a vague phrase:
For I have many eyes in my service, and I believe that you know where this precious thing now lies. Is it not so?
At that point in time the finding of the Ring was known only to Gandalf, Frodo, Bilbo, Gollum and Sauron. None of them either had contact with Saruman or had informed him. IAFAIK the source of his knowledge is explained nowhere.
Some reflections from Megan N. Fontenot on various versions of that part of the story seem relevant (quoting):
Christopher also points out that in version C, Saruman knows of the dream that visited Faramir and Boromir, and of Boromir’s journey to Rivendell. He also guesses, based on the words of the dream, that the Ring itself is on the way thither in the hands of a Halfling (UT 362). It’s not clear how or why Saruman had this private knowledge, though it is said later, in The Treason of Isengard, that Saruman and Denethor had likely been in some contact via the palantíri (462).
It is risky to search for explanations in earlier versions dismissed by the very author, of course. Yet, a contact Saruman-Denethor via the palantiri might potentially explain Saruman's suspicions. Saruman would be better placed than Denethor to understand the meaning of these words:
Seek for the Sword that was broken:
In Imladris it dwells;
There shall be counsels taken
Stronger than Morgul-spells.
There shall be shown a token
That Doom is near at hand,
For Isildur’s Bane shall waken,
And the Halfling forth shall stand.
Any thoughts on this to share: very welcome.