One of the major challenges in the processing of hydrogels based on poly(ethylene glycol) (PEG) is to create mechanically robust electrospun hydrogel scaffolds without chemical crosslinking postprocessing. In this study, this is achieved by the introduction of physical crosslinks in the form of supramolecular hydrogen bonding ureido-pyrimidinone (UPy) moieties, resulting in chain-extended UPy-PEG polymers (CE-UPy-PEG) that can be electrospun from organic solvent. The resultant fibrous meshes are swollen in contact with water and form mechanically stable, elastic hydrogels, while the fibrous morphology remains intact. Mixing up to 30 wt% gelatin with these CE-UPy-PEG polymers introduce bioactivity into these scaffolds, without affecting the mechanical properties. Manipulating the electrospinning parameters results in meshes with either small or large fiber diameters, i.e., 0.63 ± 0.36 and 2.14 ± 0.63 µm, respectively. In that order, these meshes provide support for renal epithelial monolayer formation or a niche for the culture of cardiac progenitor cells.