Steady axisymmetric laminar boundary-layer slip flow of a viscous incompressible fluid and heat transfer towards a vertical stretching cylinder in the presence of a uniform magnetic field is investigated. It is assumed that the left surface of the cylinder is heated by a hot convective flow. Using a similarity transformation, the governing system of partial differential equations is first transformed into a system of coupled nonlinear ordinary differential equations. The resulting intricate nonlinear boundary value problem is solved numerically by the fourth-order Runge – Kutta method with the shooting iteration technique. The analytical solutions are presented for a special case. The effects of various physical parameters on the velocity and temperature profiles are discussed through graphs. The values of the skin friction coefficient and the Nusselt number are tabulated and examined. It is found that the thermal boundary layer thickness increases with an increase in the velocity slip, the magnetic field, the surface convection parameter and the curvature parameter and decreases with the Prandtl number.