In this work, we propose an inexact interior proximal-type algorithm for solving convex second-order cone programs. This kind of problem consists of minimizing a convex function (possibly nonsmooth) over the intersection of an affine linear space with the Cartesian product of second-order cones. The proposed algorithm uses a variable metric, which is induced by a class of positive-definite matrices and an appropriate choice of regularization parameter. This choice ensures the well definedness of the proximal algorithm and forces the iterates to belong to the interior of the feasible set. Also, under suitable assumptions, it is proven that each limit point of the sequence generated by the algorithm solves the problem. Finally, computational results applied to structural optimization and support vector machines are presented.