<p>We establish the existence and stability of transonic shock solutions to the three-dimensional axisymmetric Euler system with an external force in a cylinder under perturbations of the incoming supersonic flow, the exit pressure, the external force and the nozzle wall. The external force has a stabilization effect on the transonic shock in the straight cylinder and the shock position is uniquely determined. The main difficulties for the axisymmetric flows are the corner singularities near the intersection of the shock surface and the nozzle wall and the singularity along the symmetry axis. One of the key ingredients in the analysis is the introduction of a modified Lagrangian transformation that is invertible near the axis and straightens the streamlines. Another one is an equivalent reformulation of the Rankine-Hugoniot conditions so that the shock front is uniquely determined by an algebraic equation.</p>