The tunneling site of the electron in a molecule exposed to a strong laser field determines the initial position of the ionizing electron and, as a result, has a large impact on the subsequent ultrafast electron dynamics on the polyatomic Coulomb potential. Here, the tunneling site of the electron of H⁺₂ ionized by a strong circularly polarized (CP) laser pulse is studied by numerically solving the time-dependent Schrödinger equation. We show that the electron removed from the down-field site is directly driven away by the CP field and the lateral photoelectron momentum distribution (LPMD) exhibits a Gaussian-like distribution, whereas the corresponding LPMD of the electron removed from the up-field site differs from the Gaussian shape due to the Coulomb focusing and scattering by the down-field core. Our current study presents the direct evidence clarifying a long-standing controversy over the tunneling site in H⁺₂ and raises the important role of the tunneling site in strong-field molecular ionization.