The estimates made by Mathews et al. <2002> for the strengths of magnetic fields at the core mantle boundary and the inner core boundary were based on the not unreasonable hypothesis that the torques generated by differential wobbles between the fluid core on the one hand and the mantle and the inner core on the other are due to the magnetic fields crossing the core mantle and inner core boundaries. In this paper, we investigate the possibility that viscous drag at these boundaries might contribute significantly to the torques. Our objectives are to find out, first, whether it would become possible, with part of the torques accounted for by viscosity, to reduce the estimates for the magnetic field strengths at the two boundaries from the values obtained by Mathews et al. to the considerably lower values that had been favored earlier and, second, whether any bounds can be placed on the fluid viscosity itself. For this purpose, we have generalized the theory of Buffett et al. <2002> to obtain new expressions for the coupling constants representing the torques at the two boundaries which include the effects due to both electromagnetism and viscosity. With the use of these we find that Mathews et al.'s estimates for the coupling constants (which are independent of the mechanism of the coupling) place upper bounds on the viscosities at the core mantle and inner core boundaries. These bounds rule out not only the very high viscosity called for in recent literature relating to the translational oscillation modes of the inner core but also other earlier estimates, with the exception of those derived from theoretical and experimental studies on liquid metals. We find also that the magnetic field strength at the inner core boundary cannot be brought down to anywhere near the levels deduced from geodynamo theories unless the inner core ellipticity and/or the density jump at the inner core boundary are assigned values that appear unrealistic. Our theoretical formulation takes the variations of the unperturbed magnetic field and the ambient flow in the fluid core to be on spatial scales that are large compared to the kilometer-scale thickness of the boundary layers which dominate the coupling of the fluid core to the mantle and the inner core at the tidal frequencies relevant to nutations. Possible variations on scales of a few kilometers or less are not taken into account. Such fine-scale maps of the fields are not obtainable, at least at present, and so definitive computations are not possible. Our assessment, however, is that the effects of such small-scale variations on the torques over the boundaries of the fluid core are unlikely to be large and that their impact on our conclusions is likely to be not significant.