Abstract A remarkable property of cellulose-based materials is that they can absorb huge amounts of water (25% of the dry mass) from ambient vapor, in the form of bound water confined at a nanoscale in the amorphous regions of the cellulose structure. The control of the dynamics of sorption and desorption of bound water is a major stake for the reduction of energy consumption and material or structure damages, but in the absence of direct observations this process is still poorly known. Here we present measurements of bound water transport thanks to Nuclear Magnetic Resonance relaxometry and Magnetic Resonance Imaging measurements. We show that the bound water is transported along the fibers and throughout the network of fibers in contact. For each material a single transport diffusion coefficient value allows to represent the processes over the whole range of saturation. The dependence of the transport diffusion coefficient on the fiber density and orientation is then analyzed to deduce the (elementary) transport diffusion coefficient of bound water along a cellulose fiber axis. This constitutes fundamental physical data which may be compared with molecular simulations, and opens the way to the prediction and control of sorption dynamics of all cellulosic materials or other hygroscopic materials.