Composite biocompatible scaffolds, obtained using the electrospinning (ES) technique, are highly promising for biomedical application thanks to their high surface area, porosity, adjustable fiber diameter, and permeability. However, the combination of synthetic biodegradable (such as poly(ε-caprolactone) PCL) and natural (such as gelatin Gt) polymers is complicated by the problem of low compatibility of the components. Previously, this problem was solved by PCL grafting and/or Gt cross-linking after ES molding. In the present study, composite fibrous scaffolds consisting of PCL and Gt were fabricated by the electrospinning (ES) method using non-functionalized PCL1 or NHS-functionalized PCL2 and hexafluoroisopropanol as a solvent. To provide covalent binding between PCL2 and Gt macromolecules, NHS-functionalized methyl glutarate was synthesized and studied in model reactions with components of spinning solution. It was found that selective formation of amide bonds, which provide complete covalent bonding of Gt in PCL/Gt composite, requires the presence of weak acid. With the use of the optimized ES method, fibrous mats with different PCL/Gt ratios were prepared. The sample morphology (SEM), hydrolytic resistance (FT-IR), cell adhesion and viability (MTT assay), cell penetration (fluorescent microscopy), and mechanical characteristics of the samples were studied. PCL2-based films with a Gt content of 20 wt% have demonstrated the best set of properties.