We investigate avascular tumour growth as a two-phase process consisting of cells and liquid. Initially, we simulate a continuum moving-boundary model formulated by Byrne, King, McElwain, Preziosi, (Applied Mathematics Letters, 2003, 16, 567-573) in one dimension and analyse the dependence of the tumour growth on the natural nutrient and cell concentration levels outside of the tumour along with its ability to model known biological dynamics of tumour growth. We investigate linear stability of time-dependent solution profiles in the moving-boundary formulation of a limit case (with negligible nutrient consumption and cell drag) and compare analytical predictions of their saturation, growth and exponential decay against numerical simulations of the full one dimensional model formulated in the article cited above. With this limit case and its time-dependent solution, we analytically obtained a critical nutrient concentration that determines whether a tumour will grow or decay. Then, we formulated the analogous model and boundary conditions for tumour growth in two dimensions. By considering the same limit case and its time-dependent solutions in two dimensions, we obtain an asymptotic limit of the two-dimensional perturbations for large time in the case where the tumour is growing by using the method of matched asymptotic approximations. Having characterised an asymptotic limit of the perturbations, we compare it to its numerical counterpart and to the time-dependent solution profiles in order to analytically obtain a condition for instability.