We study a recently introduced two-person combinatorial game, the $(a,b)$-monochromatic clique transversal game which is played by Alice and Bob on a graph $G$. As we observe, this game is equivalent to the $(b,a)$-biased Maker-Breaker game played on the clique-hypergraph of $G$. Our main results concern the threshold bias $a_1(G)$ that is the smallest integer $a$ such that Alice can win in the $(a,1)$-monochromatic clique transversal game on $G$ if she is the first to play. Among other results, we determine the possible values of $a_1(G)$ for the disjoint union of graphs, prove a formula for $a_1(G)$ if $G$ is triangle-free, and obtain the exact values of $a_1(C_n \,\square\, C_m)$, $a_1(C_n \,\square\, P_m)$, and $a_1(P_n \,\square\, P_m)$ for all possible pairs $(n,m)$.