Let K(X,Y) be the space of compact operators. For a proximinal subspace Z⊂Y, this paper deals with the question, when does every Y-valued compact operator admit a Z-valued compact best approximation? For any reflexive Banach space X and for a L1-predual space Y, if Z⊂Y is a strongly proximinal subspace of finite codimension, we show that K(X,Z) is a proximinal subspace of K(X,Y) under an additional condition on the position of K(X,Z). When Y is a c0-direct sum of finite dimensional spaces we achieve a strong transitivity result by showing that for any proximinal subspace of finite codimension Z⊂Y, every Y-valued bounded operator admits a best Z-valued compact approximation. © 2021 Elsevier Inc.