We use the framework of asymptotic safety above the Planck scale to constrain the parameter space of simple models of new physics that can accommodate the measured value of the anomalous magnetic moment of the muon and the relic density of dark matter. The models we consider comprise an inert scalar field and two vector-like colorless fermions that communicate to the muons through Yukawa-type interactions. The presence of an interactive UV fixed point in the system of gauge and Yukawa couplings imposes a set of boundary conditions at the Planck scale, which allow one to derive unique phenomenological predictions in each case. We additionally apply to the models constraints from the measured branching ratio of h->mu mu, direct LHC searches for electroweak production with leptons and missing energy in the final state, and the dark matter relic density. We find that they further restrict the available parameter space.