The Newton method for variational inequality problem is locally and quadratically convergent. By using a differentiable merit function, Taji, Fukushima and Ibaraki[1] have given a globally convergent modified Newton method for the strongly monotone variational inequality problem and proved their method to be quadratically convergent under some additional assumptions. In this paper we propose to present a trust region-type modification of Newton method for the strictly monotone variational inequality problem using the same merit function as that in [1]. It is then shown that our method is well defined and globally convergent and that,under the same assumptions as those in [1], our algorithm reduces to the basic Newton method and hence the rate of convergence is quadratic. Computational experimence indicates the efficiency of the proposed method.