Preheat oven to 350 degrees and spray a bundt cake pan with cooking spray.
In one bowl, beat butter on medium speed with an electric mixer until creamy and smooth. Add half of cake mix and buttermilk. Mix until incorporated. Add second half of cake mix, egg whites, and vanilla extract. Mix well. Set aside.
In another bowl, beat butter on medium speed with an electric mixer until creamy and smooth. Add half of cake mix and buttermilk. Mix until incorporated. Add second half of cake mix and eggs. Mix well.
Add a layer of red velvet cake batter to the prepared pan. Top it with a layer of white cake batter, using a spatula to spread it out evenly. Repeat with another layer of red velvet and white cake batter. Insert a wooden pick into the batter and run through the center of the pan.
Bake for 40-45 minutes, or until center of cake is set and a toothpick inserted in the middle comes out clean. Let cool for 10-15 minutes, then turn out on a cake plate. Cool completely before frosting.