In this paper a Nitsche-type discretization based on a discontinuous Galerkin (DG) method for an elliptic two-dimensional problem with discontinuous coefficients is considered. The problem is posed on a polygonal region $\Omega$ which is a union of $N$ disjoint polygonal subdomains $\Omega_i$ of diameter $O(H_i)$. The discontinuities of the coefficients, possibly very large, are assumed to occur only across the subdomain interfaces $\partial \Omega_i$. Inside each subdomain, a conforming finite element space on a quasi-uniform triangulation with mesh size $O(h_i)$ is introduced. To handle the nonmatching meshes across the subdomain interfaces, a DG discretization is applied only on the interfaces. For solving the resulting discrete system, a FETI--DP-type method is proposed and analyzed. It is established that the condition number of the preconditioned linear system is estimated by $C(1 + \max_i \log H_i/h_i)^2$ with a constant $C$ independent of $h_i$, $h_i/h_j$, $H_i$ and the jumps of coefficients. The method is well suited for parallel computations and it can be extended to three-dimensional problems. Numerical results are presented to validate the theory.