Strong-field ionization of atoms can be investigated on the attosecond time scale by using the attoclock method, i.e. by observing the peak of the photoelectron momentum distribution (PMD) after applying a laser pulse with a two-dimensional polarization form. Examples for such laser fields are close-to-circular or bicircular fields. Here, we report numerical solutions of the time-dependent Schrödinger equation for bicircular fields and a comparison with a compact classical model to demonstrate that the tunnel-exit position, i.e. the position where the electron emerges after tunnel ionization, is encoded in the PMD. We find that the tunnel-exit position depends on the transverse velocity of the tunneling electron. This gives rise to a momentum-dependent attoclock shift, meaning that the momentum shift due to the Coulomb force on the outgoing electron depends on which slice of the momentum distribution is analysed. Our finding is supported by a momentum-space-based implementation of the classical backpropagation method.