The internalization of matter by phagocytosis is of key importance in the defence against bacterial pathogens and in the control of cancerous tumour growth. Despite the fact that phagocytosis is an inherently mechanical process, little is known about the forces and energies that a cell requires for internalization. Here, we use functionalized magnetic particles as phagocytic targets and track their motion while actuating them in an oscillating magnetic field, in order to measure the translational and rotational stiffnesses of the phagocytic cup as a function of time. The measured evolution of stiffness reveals a characteristic pattern with a pronounced peak preceding the finalization of uptake. The measured stiffness values and their time dependence can be interpreted with a model that describes the phagocytic cup as a prestressed membrane connected to an elastically deformable actin cortex. In the context of this model, the stiffness peak is a direct manifestation of a previously described mechanical bottleneck, and a comparison of model and data suggests that the membrane advances around the particle at a speed of about 20 nm s-1. This approach is a novel way of measuring the progression of emerging phagocytic cups and their mechanical properties in situ and in real time.