We interpolate a new family of Teichmuller spaces T-#(X) between the universal Teichmuller space T and its little subspace T-0. Each T-#(X) is defined by prescribing a subset X of the unit circle as the exceptional set of the vanishing property for T-0. The inclusion relation of X induces a natural inclusion of T-#(X), and an approximation of T by an increasing sequence of T-#(X) is investigated. In this paper, we discuss the fundamental properties of T-#(X) from the viewpoint of the quasiconformal theory of Teichmuller spaces. We also consider the quotient space of T by T-#(X) as an analog of the asymptotic Teichmuller space.