In this work, we present a systematic study on the feasibility of probing the largely unexplored transverse-momentum-dependent gluon Sivers function (GSF) in open charm production, and high-pT charged dihadron and dijet production at a future high-energy, high-luminosity electron-ion collider (EIC). The Sivers function is a measure for the anisotropy of the parton distributions in momentum space inside a transversely polarized nucleon. It is proposed that it can be studied through single spin asymmetries in the photon-gluon fusion subprocess in electron-proton collisions at the EIC. Using a well-tuned Monte Carlo model for deep inelastic scattering, we estimate the possible constraints of the GSF from the future EIC data. A comparison of all the accessible measurements illustrates that the dijet channel is the most promising way to constrain the magnitude of the GSF over a wide kinematic range.