In a large mixing bowl, combine shredded chicken, taco seasoning, sour cream, cheddar cheese, and chopped green chilies. Stir well until all ingredients are evenly incorporated and the mixture is creamy.

In a saucepan over medium-high heat, melt the cubed Velveeta cheese along with the undrained diced tomatoes with green chilies. Stir continuously until the cheese is completely melted and the sauce is smooth and well combined.

Lay the flour tortillas flat on a clean surface. Spoon about ½ to ¾ cup of the chicken filling into the center of each tortilla, then roll them tightly like burritos to enclose the filling.

Place the rolled tortillas seam side down, side by side, in a baking casserole dish, making sure they fit snugly.

Pour the warm queso sauce evenly over the assembled tortillas, covering them completely.

Bake in a preheated oven at 350°F (175°C) for 20 to 25 minutes, or until the sauce is bubbly and the enchiladas are heated through.