We present calculations of magnetic potential functions associated with the perturbation of Saturn's planetary magnetic field by a rotating, equatorially situated disc of plasma. Such structures are central to the dynamics of the rapidly rotating magnetospheres of Saturn and Jupiter. They are ‘fed’ internally by sources of plasma from moons such as Enceladus (Saturn) and Io (Jupiter). For these models, we use a scaled form of Caudal's Euler potentials for the Jovian magnetodisc field. In this formalism, the magnetic field is assumed to be azimuthally symmetric about the planet's axis of rotation, and plasma temperature is constant along a field line. We perturb the dipole potential (‘homogeneous’ solution) by using simplified distributions of plasma pressure and angular velocity for both planets, based on observations by the Cassini (Saturn) and Voyager (Jupiter) spacecraft. Our results quantify the degree of radial ‘stretching’ exerted on the dipolar field lines through the plasma's rotational motion and pressure. A simplified version of the field model, the ‘homogeneous disc’, can be used to easily estimate the distance of transition in the outer magnetosphere between pressure-dominated and centrifugally dominated disc structure. We comment on the degree of equatorial confinement as represented by the scaleheight associated with disc ions of varying mass and temperature. For the case of Saturn, we identify the principal forces which contribute to the magnetodisc current and make comparisons between the field structure predicted by the model and magnetic field measurements from the Cassini spacecraft. For the case of Jupiter, we reproduce Caudal's original calculation in order to validate our model implementation. We also show that compared to Saturn, where plasma pressure gradient is, on average, weaker than centrifugal force, the outer plasma disc of Jupiter is clearly a pressure-dominated structure.