Proper processing of bulk metallic glasses (BMGs) under pressure is a promising approach to tailor their properties. However, to fully understand how pressure processing affect the final glass properties, a clearer understanding of how the pressure affects the structure of the glass at both short- and medium range levels is required. Accordingly, using molecular dynamics simulations, we study the effect of cooling under pressure on the local structure and the medium-range connectivity in a model Tantalum monatomic metallic glass. Crystalline grains form in the Ta sample with increasing the pressure under which the sample was cooled. These observations were confirmed by decreasing the fivefold symmetry with increasing pressure. The connectivity type between the perfect icosahedra was determined and showed that when cooled under pressure, intercross sharing is favored in the higher pressure. This work gives insights into understanding local structural changes induced by the pressure in metallic glasses.