A number of contributions have been made to model thermal contact problems involving perfect/imperfect thermal interfaces. The Fourier-, Cattaneo-, and Jeffreys-types of thermal flux modes have been exploited to govern the heat transfer processes of adjacent regions. However, most of the existing studies consider using only one type of these thermal flux models to describe the thermal physics of the entire system such that the mathematical model may fail to precisely describe the physics within the multi-domain system, especially, when the mean free paths of different adjacent domains span in a large range with respect to the characteristic dimension of the material. To circumvent this issue, a generalized heat conduction model, termed C- and F-processes heat conduction model, with the consideration of perfect/imperfect thermal interface is proposed and formulated under the expression of a first-order time dependent differential-algebraic equation in which the interface conditions are treated as algebraic equations. The unified time integration of GSSSS-1 is extended to solve the resulting differential-algebraic system with Index 2 constraints. The numerical results illustrate that the proposed framework has a better capacity of describing the thermal contacts with/without the thermal resistance of Fourier-Fourier, Fourier-Cattaneo, and other general cases. (C) 2017 Published by Elsevier Ltd.