Fluid-particle transport systems are highly relevant in various engineering applications, including oil sands mining and polymer processing. In many scenarios, it is critical to account for the viscoelastic nature of the fluid or the susceptibility of particles to magnetic permeability. To address these challenges, a novel numerical algorithm has been developed using an open-source finite-volume viscoelastic fluid flow solver coupled with an immersed boundary method. This algorithm is capable of simulating the behaviour of rigid magnetic spherical particles subjected to an external magnetic field. The accuracy and efficacy of the algorithm were assessed through a series of benchmark simulations, including the sedimentation of a sphere in a confined domain, the rotation of a sphere in a simple shear flow, cross-stream migration of a neutrally buoyant sphere in steady Poiseuille flow, sedimentation of two spheres exhibiting drafting, kissing, and tumbling (DKT) dynamics with and without an external magnetic field, and the movement of random arrays of magnetic spheres.