  We introduce the concept of Steinness index related to the Stein neighborhood
basis. We then show several results: (1) The existence of Steinness index is
equivalent to that of strong Stein neighborhood basis. (2) On the
Diederich-Forn{\ae}ss worm domains in particular, we present an explicit
formula relating the Steinness index to the well-known Diederich-Forn{\ae}ss
index. (3) The Steinness index is 1 if a smoothly bounded pseudoconvex domain
admits finitely many boundary points of infinite type.
