The book describes how, after the initial rise to power of the Corleone family post-Fanucci, the more clever Tessio was given a "free hand" in Brooklyn, while the loyal (but less savvy) Clemenza was yoked to the Corleone Family more closely. Tessio was much more accustomed to acting semi-autonomously, independently of the Don's control.

In my opinion, Clemenza and Tessio were Vito's pet dog and cat respectively. Clemnza the dog: loyal, strong, friendly, but subservient and in need of his master's direction. Tessio the cat: sly, clever, independent, free to roam with a more ambiguous and tenuous degree of loyalty.

So I don't think that the "dog" would ever have followed the "cat" and moved against the Don. A dog never abandons its master, whereas a cat will always act in its own interests.

If Tessio and Barzini had succeeded, they would probably have cut a deal with Clemenza. Clemenza would have reluctantly accepted the change in leadership, but not approved of it. No sense in whacking Clemenza unless they had a capable successor as caporegime.