The present work represents a progress towards the understanding of the mechanical behaviour of the fibre/matrix interface during push-in tests of fibre-reinforced polymer-matrix composites. Finite element simulations incorporating a cohesive zone model are used for this purpose. Different values of interface strength, interface fracture toughness, fibre diameter and friction coefficient are considered to study how they affect the load-displacement curves. A critical value of the displacement exists, being independent of the fibre diameter for given values of interface strength and fracture toughness, marking the separation between two regimes: (i) a cohesive-dominated zone interaction and (ii) a frictional contact between debonded fibre and matrix. Maps showing the different regimes are constructed, proving their helpfulness to tune the mechanical properties of the interface in order to favour a certain mechanical response. Finally, we study the debonding velocity and how this is affected by the mechanical properties of the interface providing an empirical relation.