Normal-to-malignant transformation is a poorly understood process associated with cellular biomechanical properties. These are strongly dependent on the dynamical behaviour of water, known to play a fundamental role in normal cellular activity and in the maintenance of the three-dimensional architecture of the tissue and the functional state of biopolymers. In this study, quasi-elastic neutron scattering was used to probe the dynamical behaviour of water in human cancer specimens and their respective surrounding normal tissue from breast and tongue, as an innovative approach for identifying particular features of malignancy. This methodology has been successfully used by the authors in human cells and was the first study of human tissues by neutron scattering techniques. A larger flexibility was observed for breast versus tongue tissues. Additionally, different dynamics were found for malignant and non-malignant specimens, depending on the tissue: higher plasticity for breast invasive cancer versus the normal, and an opposite effect for tongue. The data were interpreted in the light of two different water populations within the samples: one displaying bulk-like dynamics (extracellular and intracellular/cytoplasmic) and another with constrained flexibility (extracellular/interstitial and intracellular/hydration layers).