In magnetic fluids the viscosity can depend on an external magnetic field. We theoretically investigate the influence of this magneto-viscous effect on the thermal convection thresholds for viscoelastic fer-rofluids, which are described by a linear Oldroyd model. Such a system is influenced by a static magnetic field not only via the Kelvin force, but also through the magneto-viscous effect. In particular, we find that these two contributions compete oppositely when the threshold for the oscillatory instability is considered. While the Kelvin force tends to decrease the critical Rayleigh number, the magneto-viscous effect increases it. The critical properties at the onset of the oscillatory instability are discussed as a function of the viscoelastic parameters, the external field strength, and the magneto-viscous coefficient. The transition between the stationary and the oscillatory instability is only slightly affected by the magneto-viscous effect. Examples for codimension-2 lines are given. (C) 2017 Elsevier B.V. All rights reserved.