In this work, a novel polyurethane (PU) system based on poly(ethylene-butylene) adipate diol, 1,6-hexamethylene diisocyanate, 1,4-butanediol, and ascorbic acid was used to prepare scaffolds with potential applications in bone tissue engineering. Two fabrication methods to obtain porous materials were chosen: phase separation (PS)/salt particle leaching (PL) and solvent casting (SC)/salt PL. The calculated porosity demonstrated that scaffolds with a higher porosity were obtained (76–86%) using the PS/PL method. The morphology of pores was examined with the use of optical stereomicroscope and scanning electron microscope. The appropriate porosity, pore morphology, and swelling properties for bone tissue scaffolds were obtained for our novel PU system by the PS/PL method using 15 wt% solution of PU in the mixture of dimethylformamide (DMF)/tetrahydrofuran solvents and by SC/PL method from 20 wt% from DMF. The distilled water and saline water swelling for those samples was also appropriate for bone tissue scaffold application.