A flexible and dependable method that has been extensively employed to construct nanofibrous scaffolds that resemble the extracellular matrix made from polymeric materials is electrospinning (ES). ES is superior to other techniques because of its unique capacity to create nanofibers with a high surface-to-volume ratio, low cost, simplicity of setup, freedom in material choice, and ability to alter the surface attributes and usefulness of the nanofibers. However, the low productivity of nanofibrous membrane from conventional ES with the generation of tightly packed nanofibrous sheet-like two-dimensional membranes impedes cellular infiltration into scaffolds during tissue regeneration. Moreover, toxic organic solvents are desired for polymer dissolution for ES. Such solvents produce volatile organic compounds (VOCs) during electrospinning, which can degrade the indoor air quality of working place. Furthermore, when electrospun membranes containing traces of such VOCs are employed as tissue scaffolds, it may cause serious effect to cells and tissue. This justifies the need for alternative green solvents which are not only environmentally friendly, non-toxic, and low-cost but also biocompatible with medicinal values. Therefore, this review mainly focuses on summarizing the recent advances in ES machines, fabrication of three-dimensional (3D) spongy nanofibrous membrane, and introducing green solvent for polymer processing. Finally, based on the findings of the existing literature and our experience, this review mainly focuses on essential oils as future “greener” alternatives to current toxic solvents used in ES process.