The 21 cm-galaxy cross-power spectrum is expected to be one of the promising probes of the Epoch of Reionization (EoR), as it could offer information about the progress of reionization and the typical scale of ionized regions at different redshifts. With upcoming observations of 21 cm emission from the EoR with the Low Frequency Array (LOFAR), and of high-redshift Ly α emitters with Subaru's Hyper Suprime-Cam (HSC), we investigate the observability of such cross-power spectrum with these two instruments, which are both planning to observe the ELAIS-N1 field at z = 6.6. In this paper, we use N-body + radiative transfer (both for continuum and Ly α photons) simulations at redshift 6.68, 7.06 and 7.3 to compute the 3D theoretical 21 cm-galaxy cross-power spectrum and cross-correlation function, as well as to predict the 2D 21 cm-galaxy cross-power spectrum and cross-correlation function expected to be observed by LOFAR and HSC. Once noise and projection effects are accounted for, our predictions of the 21 cm-galaxy cross-power spectrum show clear anti-correlation on scales larger than ˜60 h-1 Mpc (corresponding to k ˜ 0.1 h Mpc-1), with levels of significance p = 0.003 at z = 6.6 and p = 0.08 at z = 7.3. On smaller scales, instead, the signal is completely contaminated. On the other hand, our 21 cm-galaxy cross-correlation function is strongly contaminated by noise on all scales, since the noise is no longer being separated by its k modes.