Preheat the oven to 350°. Combine the shredded cheese, tomatoes and green chilies, sour cream, and taco seasoning in a mixing bowl. Stir well. Set aside.
Pour the enchilada sauce in a casserole dish. Spread to coat the entire bottom.
Lay the enchiladas on the counter. Divide the filling among the enchiladas. Roll up and place the rolled enchiladas on top of the enchilada sauce.
Heat together the cubed Velveeta, milk, and green chilies. Stir until the cheese is melted. Pour the cheese sauce evenly down the center of the casserole.
Bake for 20-30 minutes, or the sauce is bubbly around the edges.