In this study, micro-size copper particles (less than 25 μm) were incorporated into polyurethane (PU) using a solution mixing method and spin-coating technique to fabricate composite films in concentrations from 0.5 to 20 vol.%. The conductivity behaviour of these composites under pressure was studied experimentally and numerically. The conductivity measurements were performed in-plane and through-thickness under pressure. It was found that changes in conductivity only occurred in the z-direction under an applied pressure from 1 to 20 kPa. The results showed that pressure could induce conductivity up to about 7.2 × 10−1 S∙m−1 for composites with a Cu concentration higher than 2.6 vol.%. It seems that applied pressure reduced the thickness of the polymer film, decreasing the distance between copper particles and promoting the formation of a conductive network, thus making the material conductive. A semi-analytical model that can accurately provide the percolation threshold (PT) concentration was used to fit the experimental conductivity. The PT concentrations for PU-Cu composite ranged from 7.1 vol.% to 1.4 vol.% and decreased with the rise in pressure. This is known as a pressure-induced percolation transition phenomenon (PIPT). Finally, the finite element method based on the representative volume element model (FE-RVE) simulation technique was used to predict the conductivity behaviour. This numerical simulation provided a good description of the experimental conductivity after the PT and correctly predicted the PT concentration. This study shows that FE-RVE could be used to effectively simulate the influence of pressure on the electrical properties of a polymer–metal composite, reducing the need for costly and time-consuming experiments.