At the Council of Elrond Legolas reveals that Gollum has escaped during a well-planned Orc attack. According to Legolas the Orcs were from over the mountains and unused to the woods but they headed back via Dol Guldur and the Wood-Elves were afraid to follow them. The unspoken assumption was that Sauron was behind it but that route makes more sense if they were heading to Isengard. Sauron had already personally interrogated Gollum about the Ring whereas Saruman may have thought Gollum worth questioning.
Unlikely; "the mountains" that Legolas mentions are most probably the Mountains of Mirkwood, and he says later in the same passage that they had tracked the Orcs close to Dol Guldur:
We have failed to recapture Gollum. We came on his trail among those of many Orcs, and it plunged deep into the Forest, going south. But ere long it escaped our skill, and we dared not continue the hunt; for we were drawing nigh to Dol Guldur, and that is still a very evil place; we do not go that way.
There doesn't seem to be any evidence that Saruman was even aware of Gollum, and it's getting too close to "what do you think of my theory?" territory to go any further.
I've always thought that there are some canonicity issues with 'The Hunt for the Ring' (Unfinished Tales), but if we accept this material then the answer is 'no'. From version A:
So it was that Sauron prepared two strokes ... The orcs assailed the realm of Thranduil with orders to recapture Gollum ...
and from version B:
[Sauron] attacked Thranduil and Gondor at about the same time. He had these two additional objects: to capture or kill Gollum, or at least to deprive his enemies of him...
Finally, the route toward Dol Guldur makes the most sense if the orcs were in fact heading for Dol Guldur.
I think you've looked at all the text there is on this question; but I don't see that the text allows us to come to any kind of decision on this. There's no discussion anywhere in the book of how Saruman may have come to know of Gollum's existence, or his whereabouts; nor of why he would have considered Gollum worth questioning if he already felt he had power and a ring of his own. Saruman already knew where the Ring was, or close enough.
Sauron was behind the attack, and it was directed by the Nazgûl
In Tolkien's The Hunt for the Rings manuscripts, he says that this attack was orchestrated by Sauron, with one or more of the Nazgûl present on scene to direct it.
Sauron had already interrogated Gollum, but the goal here was to make sure no one else did the same. Sauron discovered that Gandalf was visiting Gollum and he did not want Gandalf to be able to pry any information about the ring from him.
Several of the Nazgûl must remain in A[nduin] Vale. One or more actually direct the attack on Thranduil when Gollum escapes. Sauron thinks it vital to have him captured again and/or killed.
The Hunt for the Ring. Marquette MSS 4/2/35, quoted in The Lord of the Rings: A Reader's Companion
So it was that Sauron prepared two strokes - in which many saw the beginnings of the War of the Ring. They were made together. The Orcs assailed the realm of Thranduil, with orders to recapture Gollum; and the Lord of Morgul was sent forth openly to battle against Gondor.
The Hunt for the Ring. Marquette MSS 4/2/31, quoted in Unfinished Tales
Sauron heard the disquieting news that the Wise were aware of Gollum, and that Gandalf had passed into Thranduil's realm.
Sauron must then have been filled with anger and alarm. He resolved to use the Ringwraiths as soon as he could, for speed rather than secrecy was now important. Hoping to alarm his enemies and disturb their counsels with the fear of war (which he did not intend to make for some time), he attacked Thranduil and Gondor at about the same time. He had these two additional objects: to capture or kill Gollum, or at least to deprive his enemies of him; and to force the passage of the bridge of Osgiliath, so that the Nazgûl could cross, while testing the strength of Gondor.
The Hunt for the Ring. Marquette MSS 4/2/32, quoted in Unfinished Tales