Chitosan nanofibrous membranes are prepared via an electrospinning technique and explored as potential wound healing patches. In particular, the effect of a physical or chemical crosslinking treatment on the mat morphological, mechanical, water-related, and biological properties is deeply evaluated. The use of phosphate ions (i.e., physical crosslinking) allows us to obtain smooth and highly homogenous nanofibers with an average size of 190 nm, whereas the use of ethylene glycol diglycidyl ether (i.e., chemical crosslinking) leads to rougher, partially coalesced, and bigger nanofibers with an average dimension of 270 nm. Additionally, the physically crosslinked mats show enhanced mechanical performances, as well as greater water vapour permeability and hydrophilicity, with respect to the chemically crosslinked ones. Above all, cell adhesion and cytotoxicity experiments demonstrate that the use of phosphate ions as crosslinkers significantly improves the capability of chitosan mats to promote cell viability owing to their higher biocompatibility. Moreover, tuneable drug delivery properties are achieved for the physically crosslinked mats by a simple post-processing impregnation methodology, thereby indicating the possibility to enrich the prepared membranes with unique features. The results prove that the proposed approach may lead to the preparation of cheap, biocompatible, and efficient chitosan-based nanofibers for biomedical and pharmaceutical applications.