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We prove the nonlinear stability of the asymptotic behaviour of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarized T2 -symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant Λ. This stability result generalizes the results proven in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized T2 -symmetric vacuum spacetimes. Ann. Henri Poincaré. (doi:10.1007/s00023-021-01142-0)), which focus on the Λ=0 case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for Λ=0 , the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarized T2 -symmetric vacuum solutions than those considered in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized T2 -symmetric vacuum spacetimes. Ann. Henri Poincaré. (doi:10.1007/s00023-021-01142-0)) and Fournodavlos G et al. (2020 Stable Big Bang formation for Einstein's equations: the complete sub-critical regime. Preprint. (http://arxiv.org/abs/2012.05888)). Our results establish that the areal time coordinate takes all values in (0,T0] for some T0>0 , for certain families of polarized T2 -symmetric solutions with cosmological constant. This article is part of the theme issue 'The future of mathematical cosmology, Volume 1'. [ABSTRACT FROM AUTHOR]