We investigate the quasi-bound states of a Coulomb impurity in graphene in the presence of a magnetic field. These states exhibit the dramatic and rather rare property of discrete scale invariance when the Coulomb potential is supercritical. We show using both Wentzel-Kramers-Brillouin (WKB) approximation and numerical studies that the supercritical states are converted to subcritical states as the field is increased. The local density of states is calculated and it shows direct signatures of discrete scale invariance. In a magnetic field, these signatures are gradually destroyed in a systematic way. Hence the effect that we propose can be detected via scanning tunneling microscope experiments. The range of magnetic field and energy resolution required are compatible with existing experimental setups. These experiments can be performed in a single sample by changing the field; they do not involve changing the nuclear charge.
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