Brielle Biermann is being hounded by American Express for allegedly not paying a five-figure credit card bill, and now AmEx is going after her in court.

According to new legal documents, obtained by TMZ, American Express claims Kim Zolciak's daughter failed to pay off a $12,870.25 balance.

The credit card company claims Brielle hasn't been forking over the monthly minimum payment on her Delta SkyMiles Reserve American Express Card.

In the docs, AmEx says it's demanded payment from Brielle ... but she's failed to settle the bill ... and now they're going after her for the full balance, plus court costs.

The apple apparently doesn't fall far from the tree ... Brielle's mom is also being sued for an unpaid credit card bill at Target, just one factor in her mother's ongoing struggles.

Meanwhile, as we first reported ... Brielle's adopted father, Kroy Biermann, is being sued over an unpaid credit line from a casino trip to the Bahamas.