The magnetic network (NE) observed on the solar surface harbors a sizable fraction of the total quiet Sun flux. However, its origin and maintenance are not well known. Here we investigate the contribution of internetwork (IN) magnetic fields to the NE flux. IN fields permeate the interior of supergranular cells and show large emergence rates. We use long-duration sequences of magnetograms acquired by Hinode and an automatic feature tracking algorithm to follow the evolution of NE and IN flux elements. We find that 14% of the quiet Sun (QS) flux is in the form of IN fields with little temporal variations. IN elements interact with NE patches and modify the flux budget of the NE either by adding flux (through merging processes) or by removing it (through cancellation events). Mergings appear to be dominant, so the net flux contribution of the IN is positive. The observed rate of flux transfer to the NE is 1.5 × 1024 Mx day–1 over the entire solar surface. Thus, the IN supplies as much flux as is present in the NE in only 9-13 hr. Taking into account that not all the transferred flux is incorporated into the NE, we find that the IN would be able to replace the entire NE flux in approximately 18-24 hr. This renders the IN the most important contributor to the NE, challenging the view that ephemeral regions are the main source of flux in the QS. About 40% of the total IN flux eventually ends up in the NE.