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We consider finite element discretizations of the Biot’s consolidation model in poroelasticity with MINI and stabilized P1–P1 elements. We analyze the convergence of the fully discrete model based on spatial discretization with these types of finite elements and implicit Euler method in time. We also address the issue related to the presence of non-physical oscillations in the pressure approximation for low permeabilities and/or small time steps. We show that even in 1D a Stokes-stable finite element pair fails to provide a monotone discretization for the pressure in such regimes. We then introduce a stabilization term which removes the oscillations. We present numerical results confirming the monotone behavior of the stabilized schemes.