Set a small sauce pot over medium heat. Add the oil and flour and whisk for 1-2 minutes until thick and smooth, but not dark. Whisk in the wine and bring to a simmer.
Meanwhile, heat a fondue pot on its stand in preparation for the fondue. Slowly add the cheese to the hot liquid, stirring continuously to melt. Once all the cheese is melted into the liquid, and the mixture is smooth, stir in the pesto.
Pour the cheese fondue into the fondue pot to keep it warm and serve with bread cubes, veggies and/or fruit.