Direct access into cells' interiors is essential for biomolecular delivery, gene transfection, and electrical recordings yet is challenging due to the cell membrane barrier. Recently, molecular delivery using vertical nanowires (NWs) has been demonstrated for introducing biomolecules into a large number of cells in parallel. However, the microscopic understanding of how and when the nanowires penetrate cell membranes is still lacking, and the degree to which actual membrane penetration occurs is controversial. Here we present results from a mechanical continuum model of elastic cell membrane penetration through two mechanisms, namely through "impaling" as cells land onto a bed of nanowires, and through "adhesion-mediated" penetration, which occurs as cells spread on the substrate and generate adhesion force. Our results reveal that penetration is much more effective through the adhesion mechanism, with NW geometry and cell stiffness being critically important. Stiffer cells have higher penetration efficiency, but are more sensitive to NW geometry. These results provide a guide to designing nanowires for applications in cell membrane penetration.