Abstract The present problem is the steady boundary layer flow and heat transfer of a hyperbolic tangent fluid flowing over a vertical exponentially stretching cylinder in its axial direction. After applying usual boundary layer with a suitable similarity transformation to the given partial differential equations and the boundary conditions, a system of coupled nonlinear ordinary differential equations is obtained. This system of ordinary differential equations subject to the boundary conditions is solved with the help of Runge–Kutta–Fehlberg method. The effects of the involved parameters such as Reynolds numbers, Prandtl numbers, Weissenberg numbers and the natural convection parameter are presented through the graphs. The associated physical properties on the flow and heat transfer characteristics that is the skin friction coefficient and Nusselt numbers are presented for different parameters.