The aim of this contribution is to address the convergence study of a time and space approximation scheme for an Allen-Cahn problem with constraint and per- turbed by a multiplicative noise of Itô type. The problem is set in a bounded domain of $\mathbb{R}^d$ (with ${d}$ = 2 or 3) and homogeneous Neumann boundary conditions are considered. The employed strategy consists in building a numerical scheme on a regularized version “à la Moreau-Yosida” of the constrained problem, and passing to the limit simultaneously with respect to the regularization parameter and the time and space steps, denoted respectively by ε, ∆t and h. Combining a semi-implicit Euler-Maruyama time discretization with a Two-Point Flux Approximation (TPFA) scheme for the spatial variable, one is able to prove, under the assumption ∆t = O(ε$^{2+θ}$) for a positive θ, the convergence of such a “(ε,∆t,h)” scheme towards the unique weak solution of the initial problem, a priori strongly in $L^2$(Ω; L$^2$(0, T ; L$^2$(Λ))) and a posteriori also strongly in $L^p$(0, T ; $L^2$(Ω × Λ)) for any finite p ≥ 1.