The present study investigated the possibility of obtaining 3D printed composite constructs using biomaterial-based nanocomposite inks. The biopolymeric matrix consisted of methacrylated gelatin (GelMA). Several types of nanoclay were added as the inorganic component. Our aim was to investigate the influence of clay type on the rheological behavior of ink formulations and to determine the morphological and structural properties of the resulting crosslinked hydrogel-based nanomaterials. Moreover, through the inclusion of nanoclays, our goal was to improve the printability and shape fidelity of nanocomposite scaffolds. The viscosity of all ink formulations was greater in the presence of inorganic nanoparticles as shear thinning occurred with increased shear rate. Hydrogel nanocomposites presented predominantly elastic rather than viscous behavior as the materials were crosslinked which led to improved mechanical properties. The inclusion of nanoclays in the biopolymeric matrix limited hydrogel swelling due the physical barrier effect but also because of the supplementary crosslinks induced by the clay layers. The distribution of inorganic filler within the GelMA-based hydrogels led to higher porosities as a consequence of their interaction with the biopolymeric ink. The present study could be useful for the development of soft nanomaterials foreseen for the additive manufacturing of customized implants for tissue engineering.