Let R be a ring and n be a non-negative integer. In this paper, we introduce and study the notions of Gorenstein n-weak injective and Gorenstein n-weak flat modules by using the notion of special super finitely presented modules. On an arbitrary ring, we investigate the relationships between Gorenstein n-weak injective and Gorenstein n-weak flat modules. Among other results, we prove that any R-module admits a Gorenstein n-weak injective (resp. Gorenstein n-weak flat) cover and pre-envelope. Then, we deduce that the class of Gorenstein n-weak injective (resp. Gorenstein n-weak flat) R-modules coincides with that of two-degree Gorenstein n-weak injective (resp. two-degree Gorenstein n-weak flat) R-modules.