There is actually a well known Bundt cake that is blueberry puree swirled into the cake. The original isn't chocolate, but that wouldn't take more than some cocoa to cure. And there's no law that says you can't play with other types of berries.
It's an America's Test Kitchen recipe, so this is when I would normally say, "(sorry, paywalled)". Today however, the recipe Gods are smiling upon me. ATK (Cook's Illustrated) has allowed the recipe to be published online and available to even us mere mortals who have not yet paid their subscription fee this year.
Also, the ATK video is still not paywalled, but I expect that the change quickly as it's actually from last season (they don't charge for the current season). ATK
3 cups (15 ounces) all-purpose flour
1 1/2 teaspoon baking powder
3/4 teaspoon baking soda
1 teaspoon salt
1/2 teaspoon ground cinnamon
3/4 cup buttermilk
2 teaspoons grated
lemon zest plus 3 tablespoons juice
2 teaspoons vanilla extract
3 large eggs plus 1 large yolk, room temperature
2 1/4 sticks unsalted butter, softened
2 cups sugar
3/4 cup sugar
3 tablespoons low- or no-sugar-needed fruit pectin
Pinch of salt
2 cups fresh or thawed frozen blueberries
1 teaspoon grated lemon zest plus 1 tablespoon juice
Adjust oven rack to lower-middle position and preheat to 325 degrees. Heavily spray nonstick Bundt pan with baking spray and flour.
Whisk flour, baking powder, baking soda, salt and cinnamon together in large bowl.
Whisk buttermilk, lemon zest, juice and vanilla together in medium-size bowl.
Gently whisk eggs and yolk to combine in third bowl.
Using a stand mixer fitted with a paddle, beat butter and sugar on medium-high until pale and fluffy, about 3 minutes, scraping down bowl as needed. Reduce speed to medium and beat in half of eggs until incorporated, about 15 seconds.
Repeat with remaining eggs, scraping down bowl after incorporating. Reduce speed to low and add 1/3 of flour mixture, followed by half of buttermilk mixture, mixing until just incorporated after each addition, about 5 seconds.
Repeat using half of remaining flour mixture and all of remaining buttermilk mixture. Scrape down bowl, add remaining flour mixture and mix on medium-low until batter is thoroughly combined, about 15 seconds. Remove bowl from mixer and fold batter once or twice with a rubber spatula to incorporate any remaining flour.
Cover bowl with plastic wrap and set aside while preparing filling (batter will inflate a bit.)
Filling: Whisk sugar, pectin and salt together in a small saucepan.
Process blueberries in a blender until mostly smooth, about 1 minute.
Transfer 1/4 cup puree and lemon zest to saucepan with sugar mixture and stir to thoroughly combine.
Heat sugar-blueberry mixture over medium heat until just simmering, about 3 minutes, stirring frequently to dissolve sugar and pectin.
Transfer mixture to medium-size bowl and let cool for 5 minutes. Add remaining puree and lemon juice to cooled mixture and whisk to combine. Let sit until slightly set, about 8 minutes.
Spoon half of batter into prepared pan and smooth top. Using back of spoon, create 1/2-inch-deep channel in center of batter. Spoon half of filling into channel.
Using butter knife or small offset spatula, thoroughly swirl filling into batter (there should be no large pockets of filling remaining). Repeat swirling step with remaining batter and filling.
Bake until top is golden brown and skewer inserted in center comes out clean, about 60 to 70 minutes. Let cake cool in pan on wire rack for 10 minutes, then invert cake directly onto wire rack. Allow cake to cool for at least 3 hours before serving.
That may not be exactly what you're looking for, but it's a place to start.