Three-dimensional (3D) bioprinting opens up many possibilities for tissue engineering, thanks to its ability to create a three-dimensional environment for cells like an extracellular matrix. However, the use of natural polymers such as silk fibroin in 3D bioprinting faces obstacles such as having a limited printability due to the low viscosity of such solutions. This study addresses these gaps by developing highly viscous, stable, and biocompatible silk fibroin-based inks. The addition of 2% carboxymethyl cellulose sodium and 1% sodium alginate to an aqueous solution containing 2.5 to 5% silk fibroin significantly improves the printability, stability, and mechanical properties of the printed scaffolds. It has been demonstrated that the more silk fibroin there is in bioinks, the higher their printability. To stabilize silk fibroin scaffolds in an aqueous environment, the printed structures must be treated with methanol or ethanol, ensuring the transition from the silk fibroin’s amorphous phase to beta sheets. The developed bioinks that are based on silk fibroin, alginate, and carboxymethyl cellulose demonstrate an ease of printing and a high printing quality, and have a sufficiently good biocompatibility with respect to mesenchymal stromal cells. The printed scaffolds have satisfactory mechanical characteristics. The resulting 3D-printing bioink composition can be used to create tissue-like structures.