Wineries produce considerable amounts of grape pomace, which is a readily available natural source of bioactive phenolic compounds. In this study, grape pomace was used as a substrate for the cultivation of eleven filamentous fungi (Trametes versicolor TV6, Trametes versicolor TV8, Trametes versicolor AG613, Trametes gibbosa, Phanerochaete chrysosporium, Ceriporiopsis subvermispora, Pleurotus eryngii, Ganoderma lucidum, Ganoderma resinaceum, Humicola grisea, and Rhizopus oryzae) under solid-state conditions (SSF) for 15 days with the aim of improving the recovery of the individual phenolic compounds. Twenty-one phenolic compounds were quantified and the recovery of seventeen of them (gallic acid, ellagic acid, p-hydroxybenzoic acid, syringic acid, vanillic acid, 3,4-dihydroxybenzoic acid, ferulic acid, o-coumaric acid, p-coumaric acid, epicatechin gallate, galocatechin gallate, quercetin, kaempferol, procyanidin B1, procyanidin B2, resveratrol, and ε-viniferin) were positively affected by SSF. Ellagic acid is the most recovered compound, whose content increased 8.8-fold after 15 days of biological treatment with Ceriporiopsis subvermispora compared to the untreated initial sample. Among the microorganisms tested, the fungi Pleurotus eryngii and Rhizopus oryzae proved to be the most effective in increasing the recovery of most phenolic compounds (1.1–4.5-fold). In addition, the nutrient composition (proteins, ash, fats) of grape pomace was positively affected by the biological treatments.