In this paper, the flow of a simplest Non-Newtonian fluid in a channel of varying cross-section with permeable boundaries is investigated. The paper finds its significance in understanding the flow of Bio-fluids in the ducts of varying cross section. Perturbation technique is used to solve the governing equations of the flow phenomenon. The expressions for velocity, flow rate, wall shear stress, and the pressure drop are derived. The flux is a function of external pressure, Jeffrey parameter, and the permeability parameter. Further, if our results agree with Krishna Prasad and Chandra (Proc Nat Acad Sci 60(A) III: 317–326, 1990 [1]). Mathematica software is used find the pressure, which place a vital role in varying cross sections. Shear stress increases with increasing effects of permeability and Jeffrey parameters which is observed graphically. It is noticed that shear thinning reduces the wall shear stress and this point is stresses in the paper as well. This work helps the young researcher to develop interest in the field of Bio-fluids with varying cross-section. © 2021, Springer Nature Singapore Pte Ltd.