  We study the weak $K_s$-saturation number of the Erd\H{o}s--R\'{e}nyi random
graph $\mathbbmsl{G}(n, p)$, denoted by $\mathrm{wsat}(\mathbbmsl{G}(n, p),
K_s)$, where $K_s$ is the complete graph on $s$ vertices. Kor\'{a}ndi and
Sudakov in 2017 proved that the weak $K_s$-saturation number of $K_n$ is
stable, in the sense that it remains the same after removing edges with
constant probability. In this paper, we prove that there exists a threshold for
this stability property and give upper and lower bounds on the threshold. This
generalizes the result of Kor\'{a}ndi and Sudakov. A general upper bound for
$\mathrm{wsat}(\mathbbmsl{G}(n, p), K_s)$ is also provided.
