The Neogene sedimentary record of the Mediterranean Sea holds evidence for changes in water properties and circulation. These paleoceanographic changes have been attributed to changes in the flow through the ocean gateway between the Mediterranean Sea and the Atlantic Ocean. We use an oceanic general circulation model to achieve quantitative, physics-based understanding of the effect of changes in the depth of the gateway such as may result from tectonic activity or variation in sea level. To isolate these effects we use idealized basin geometry and impose simplified atmospheric forcing. A reference experiment with present-day sill depth demonstrates that the model simulates reasonably well the main features of the present-day Mediterranean thermohaline circulation as inferred from observations and previous numerical studies. Subsequently, a series of sensitivity simulations is performed with different sill depths. The model results show that Mediterranean temperature, salinity, and thermohaline circulation depend strongly on sill depth. As the sill depth decreases, the upper overturning cell is quasi-linearly reduced in strength, while, contrary to what one would expect, the deep cell intensifies and does so in a nonlinear way. We find that a shoaling of the sill depth induces a “blocking effect” on the outflow waters, which creates a strong recirculation in the deep layers, strengthening the deep cell. Nevertheless, deep-water formation is reduced, and, as a consequence, the ventilation of the deep layers diminishes. We identify three different circulation modes of the Mediterranean thermohaline circulation depending on the sill depth: “shallow sill,” “moderate sill,” and “deep sill” modes coupled with strong, weak, and negligible blocking effects, respectively. Our results are consistent with the pre-Messinian paleoceanographic record of the Mediterranean Sea and may be useful to understanding the behavior of other land-locked basins, both extant and from the geological past.