Diffusion-controlled stepwise polymerization of a linear polymer confined in nanoscopic slits is simulated through a Monte Carlo approach. A noticeable influence of the confinement on the kinetics is found. The confinement modifies both the spatial pair distribution function and the diffusive properties of the polymers. As a consequence, the confined system can show either faster or slower reaction kinetics with respect to the bulk system, depending on the strength of intermolecular interactions. The predicted polydispersity of the polymer is in agreement with recent theories of diffusion-controlled stepwise polymerization, and can be slightly affected by the confinement.