We investigate the notion of approximative τ-compactness in Banach spaces, where τ is the norm or the weak topology. The approximative τ-compactness of Banach spaces is characterized in several ways when the subspace is of finite codimensional. A separable Banach space $X$ is shown to be Asplund if and only if X* admits a dual $CLUR$ renorming. This property is discussed in the context of quotient spaces. We prove that, under certain conditions, the property of approximative τ-compactness passes through M-ideals. Stability results for approximative τ-compactness in the spaces of Bochner integrable functions are also presented.