Dark matter could potentially manifest itself in the form of asymmetric dark stars. In this paper we entertain the possibility of probing such asymmetric bosonic dark matter stars by the use of atomic clocks. If the dark sector connects to the standard model sector via a Higgs or photon portal, the interiors of boson stars that are in a Bose-Einstein condensate state can change the values of physical constants that control the timing of atomic clock devices. Dilute asymmetric dark matter boson stars passing through the Earth can induce frequency shifts that can be observed in separated Earth-based atomic clocks. This gives the opportunity to probe a class of dark matter candidates that for the moment cannot be detected with any different conventional method.