Core–sheath electrospinning is a rapid microfabrication process for creating multilayer polymer microfibers. This paper presents a process based on core–sheath electrospinning to fabricate poly(L-lactic acid) (PLLA) microtubes with nanopores on the tube wall. The morphology of the microtubes mimics human fenestrated capillary vessels. This study investigates the effects of the viscosities of the core and the sheath solutions on the microtube outer diameter and the nanopore size. The core solution shows a dominating influence on the microtube diameter. At the same core solution viscosity level, the microtube diameter is negatively correlated to the core-to-sheath viscosity ratio. The pore size is positively correlated to the microtube diameter. Understanding the effects of solution viscosity on microtube morphology is the prerequisite for process control and microtube product development for future biomedical applications.