<p>This study aims to examine the two-dimensional boundary layer flow and heat transfer over a nonlinearly permeable stretching/shrinking cylinder at the stagnation point, utilizing water and kerosene as base fluids with single-walled and multi-walled carbon nanotubes. The analysis considers hydromagnetic effects in the flow. The partial differential equations (PDEs) governing the fluid flow model are transformed into ordinary differential equations (ODEs) through similarity transformation and are then analyzed using bvp4c solver in MATLAB software. The impacts of the magnetic field, curvature, nanoparticle volume fraction and nonlinear parameters, as well as the influence of varying these parameters on the skin friction coefficients, heat transfer rate on surface, velocity and temperature profiles are observed. The study reveals a duality of solutions within a specific range of shrinking case, whereas a unique solution is obtained for the stretching case. The impact of magnetic and nonlinear parameter increase the rate of heat transfer. It also demonstrated that single-walled carbon nanotubes and kerosene-based fluid contribute to higher skin friction coefficients and heat transfer on the surface. A response surface methodology (RSM) is conducted to develop a correlation between the heat transfer rate, which is the response and the parameter considered in this study. The findings by using RSM found that nanoparticle volume fraction positively impacts the rate of heat transfer. The heat transfer rate is estimated to be optimized at <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(-0.0144420\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>0.0144420</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>