The present paper is concerned with estimating the effective conductivity of a composite consisting of spherical particles dispersed inside a matrix in a statistically homogeneous and isotropic way. The interface between every spherical particle and the matrix is thermally imperfect and described by the general linear isotropic imperfect model resulting from the replacement of an interphase of weak thickness by an interface of zero thickness. This general model includes as extreme particular cases Kapitza's (or lowly conducting) thermal resistance model and the highly conducting thermal imperfect interface model. The fundamental solution is derived for the problem of a spherical particle embedded, via a general imperfect interface, in an infinite matrix undergoing a remote uniform intensity boundary loading. With the help of this fundamental solution, closed-form estimates for the size-dependent effective conductivity of the composite are deduced by using the dilute distribution, Mori-Tanaka, self-consistent and generalized self-consistent schemes. These results, incorporating as particular ones all the relevant estimates reported in the literature, are discussed and compared through numerical examples. (C) 2014 Elsevier Ltd. All rights reserved.