The buoyancy force (due to Archimedes) will grow in proportion to the volume of displaced fluid.

To be more precise: Buoyancy force = (Volume of displaced fluid) X (Fluid density) X (Free fall acceleration, g)

If you push down the block into the water, the buoyancy force would grow, as the block now displaces more water. The amount of change will be given by the above formula, with the volume term exchanged by the volume of the block intially outside the water.