Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a large mixing bowl, whisk together the mashed bananas, eggs, granulated sugar, brown sugar, melted butter, yogurt, and vanilla extract until well combined.
In a separate bowl, sift together the all-purpose flour, baking soda, baking powder, and salt.
Gradually add the dry ingredients to the wet ingredients, folding gently until just combined. Do not overmix.
Pour half of the banana cake batter into the prepared cake pan and spread evenly.
Drop spoonfuls of Nutella over the batter layer, then pour the remaining banana cake batter on top.
Using a knife or skewer, gently swirl the Nutella into the batter to create a marbled effect.
Bake in the preheated oven for 45 to 50 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before slicing and serving.