The development of smart and intelligent regenerative biomaterials for skeletal muscle tissue engineering is an ongoing challenge, owing to the requirement of achieving biomimetic systems able to communicate biological signals and thus promote optimal tissue regeneration. Electrospinning is a well-known technique to produce fibers that mimic the three dimensional microstructural arrangements, down to nanoscale and the properties of the extracellular matrix fibers. Natural and synthetic polymers are used in the electrospinning process; moreover, a blend of them provides composite materials that have demonstrated the potential advantage of supporting cell function and adhesion. Recently, the decellularized extracellular matrix (dECM), which is the noncellular component of tissue that retains relevant biological cues for cells, has been evaluated as a starting biomaterial to realize composite electrospun constructs. The properties of the electrospun systems can be further improved with innovative procedures of functionalization with biomolecules. Among the various approaches, great attention is devoted to the “click” concept in constructing a bioactive system, due to the modularity, orthogonality, and simplicity features of the “click” reactions. In this paper, we first provide an overview of current approaches that can be used to obtain biofunctional composite electrospun biomaterials. Finally, we propose a design of composite electrospun biomaterials suitable for skeletal muscle tissue regeneration.