A model for avascular tumour growth with white noise is presented. The model is a set of partial differential equations describing the spatio-temporal evoluations of cell concentrations based on reaction-diffusion dynamics and the law of mass conservation. Perturbations in the form of white noise are introduced to model the effects of random processes on distinct time scales. Numerical simulations in one and two space dimensions are presented. Numerical results indicate that the proposed model is a reasonable approach that may be used to examine the effects of nutrient supply in tumour growth dynamics.