An exchange bias study has been performed on core-shell iron-iron oxide (Fe-Fe3O4) nanoclusters (NCs) of sizes 11 nm and 14 nm carrying a different core to shell ratio. NCs show complicated behaviors due to competition between interfacial exchange and Zeeman energy in the presence of magnetic field during cooling. These behaviors are accompanied by the evolution of size-dependent cluster structures in the ferromagnetic-core/ferri- or antiferro-magnetic-shell. Smaller clusters have larger coercive field, exchange bias field, and vertical magnetization shift due to the greater contribution from frozen spins of shell/interfaces and magnetic frustration by the defects and voids present at the interface. These smaller clusters thus also show more dramatic changes with the training effect. Both sizes of clusters display an additional anomaly of the upper part of the hysteresis loop at 10 K under low cooling field (0.1 kOe). This anomaly decreases with number of loop cycles with same field, and disappears with large cooling field (>0.1 kOe). It may be caused by the competition between the magnetization reversal and the magnetostatic interactions.