High Faraday rotation measures in the centers of cooling-flow clusters indicate the presence of strong magnetic fields. We examine the effects of these strong fields on the propagation of radio jets emerging from the central cD galaxies of these clusters, and the deformation of the magnetic fields by the fast-propagating jets. We argue that active regions will develop around these radio jets as a result of the violent response of the strong ambient magnetic fields. The magnetic tension can act back on the jets by influencing the development of Rayleigh-Taylor and Kelvin-Helmholtz instability modes, and by exerting a nonaxisymmetric force on the jets. If the jet propagation direction is not along the magnetic field lines, then the jet will be sharply bent by the magnetic tension. Future MHD numerical simulations that will study these effects more quantitatively should be compared directly with specific clusters. If, indeed, some properties of jets expanding from cD galaxies in cooling-flow clusters will turn out to result from interaction with strong magnetic fields in the intracluster medium at the centers of these clusters, then this will strengthen the cooling-flow model, since it will support the presence of inflow.