There's no evidence of any plan that Saruman may or may not have had in the books. What is well established is that by some time before the events of Lord of the Rings he had begun to desire the Ring for himself, but likewise there's no evidence that Saruman was even aware that Frodo had the Ring and was taking it to Mordor to be destroyed. In fact in the chapter the Uruk-Hai it is suggested that he may have thought that one of Merry or Pippin had it:
'What are they wanted for?' asked several voices. 'Why alive? Do they give good sport?'
'No! I heard that one of them has got something, something that's wanted for the War, some elvish plot or other. Anyway they'll both be questioned.'
It's evident from the context that the latter speaker here is Ugluk, the commander of the Isengard Orcs.
The best account of Saruman's long-term plans is given in the foreword to Lord of the Rings where Tolkien speculates on an alternate outcome if the Ring had been used against Sauron:
Saruman, failing to get possession of the Ring, would in the confusion and treacheries of the time have found in Mordor the missing links in his own researches into Ring-lore, and before long he would have made a Great Ring of his own with which to challenge the self-styled Ruler of Middle-earth.
Everything else is speculative and opinion-based.