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Let $G=(V,E)$ be a finite undirected graph. A set $S$ of vertices in $V$ issaid to be total $k$-dominating if every vertex in $V$ is adjacent to at least$k$ vertices in $S$. The total $k$-domination number, $\gamma_{kt}(G)$, is theminimum cardinality of a total $k$-dominating set in $G$. In this work we studythe total $k$-domination number of Cartesian product of two complete graphswhich is a lower bound of the total $k$-domination number of Cartesian productof two graphs. We obtain new lower and upper bounds for the total$k$-domination number of Cartesian product of two complete graphs. Someasymptotic behaviors are obtained as a consequence of the bounds we found. Inparticular, we obtain that$\displaystyle\liminf_{n\to\infty}\frac{\gamma_{kt}(G\Box H)}{n}\leq2\,\left(\left\lceil\frac{k}{2}\right\rceil^{-1}+\left\lfloor\frac{k+4}{2}\right\rfloor^{-1}\right)^{-1}$for graphs $G,H$ with order at least $n$. We also prove that the equality isattained if and only if $k$ is even. The equality holds when $G,H$ are bothisomorphic to the complete graph, $K_n$, with $n$ vertices. Furthermore, weobtain closed formulas for the total $2$-domination number of Cartesian productof two complete graphs of whatever order. Besides, we prove that, for $k=3$,the inequality above is improvable to $\displaystyle\liminf_{n\to\infty}\gamma_{3t}(K_n\Box K_n)/n \leq 11/5$.

Total $k$-domination in Cartesian product of complete graphs