The detection of emerging contaminants in bodies of water has steadily increased in recent years, becoming a severe problem threatening human and ecosystem health. Developing new materials with adsorption properties to remove these pollutants represents an important step toward a potential solution. In this paper, a polybutylene adipate terephthalate (PBAT) nanofibrous membrane incorporating clinoptilolite zeolite was developed and its excellent performance in removing tetracycline (TC) and methylene blue (MB) from water was demonstrated. The composite membrane was prepared in two steps: firstly, a homogeneous dispersion of clinoptilolite (1 wt% respect to polymer) in a PBAT solution (12.6 wt%) was electrospun; secondly, the electrospun membrane was subjected to an acid treatment that improved its wettability through the protonation of the surface silanol groups of clinoptilolite. The resulting membrane was hydrophilic and showed higher adsorption for TC (800 mg/g) and MB (100 mg/g), using a low dose (90 mg/L) powdered zeolite. The maximum removal capacity was obtained at neutral pH, being the cation exchange reaction the main adsorption mechanism. Pseudo-second-order kinetics and Henry’s law agree well with the proposed chemisorption and the high affinity of TC and MB for the adsorbent. The material can be reused after the removal process without generating additional contamination, although losing some effectivity.