The note seems to come from a quote in the Unfinished Tales in a section titled "Other Versions of the Story" in the chapter "The Hunt for the Ring". The quote is as follows:
He went back to the Gate, and he lied, saying that he had made Gandalf confess. He did not admit that this was his own knowledge, not being aware of how much Sauron knew of his mind and heart.
Unfinished Tales: Part Three: The Third Age, IV: The Hunt for the Ring, Other Versions of the Story
To provide context for the quote. The version of the story it comes from is labelled "Version C", in which (unlike A and B) Gandalf is still a prisoner at the time of the Nazgûl arriving at the gates of Orthanc. In C, Saruman is prepared to yield to Gandalf and beg for his help as he perceives the full horror of being in service to the Dark Tower. Saruman tells the Nazgûl "that he would go and try to discover what he knew; if that were unavailing, he would deliver Gandalf up to them." However, upon is return to the top of Orthanc, Saruman discovers Gandalf has escaped.
Saruman's lie seems to come as a combination of not wanting Sauron to know that he has lost Gandalf, not wanting Sauron to know that he was aware of the whereabouts of the Shire and searching for the Ring himself, and not wanting Sauron to know that he was willing to forgo service to him in exchange for aid from Gandalf.
The above quote comes with the following note:
Earlier in this version it is said that Sauron had at this time, by means of the palantíri, at last begun to daunt Saruman, and could in any case often read his thought even when he withheld information. Thus Sauron was aware that Saruman had some guess at the place where the Ring was; and Saruman actually revealed that he had got as his prisoner Gandalf, who knew the most.
The note isn't overly informative but tells us that whatever lies Saruman tells it is most likely that through the palantíri Sauron can perceive the truth.
In fact it is, again unlike A and B, not Gríma who reveals Saruman's lies but a Dunlending
spy of Saruman's:
This Dunlending was overtaken by several of the Black Riders as they approached the Tharbad crossing. In an extremity of terror he was haled to the Witch-king and questioned. He saved his life by betraying Saruman. The Witch-king thus learned that Saruman knew well all along where the Shire was, and knew much about it, which he could and should have told to Sauron’s servants if he had been a true ally.