A set S of vertices in V  is said to be k-dominating if every vertex in \(V\setminus S\) is adjacent to at least k vertices in S. The k-domination number, \(\gamma _{k}(G)\) , is the minimum cardinality of a k-dominating set in G. In this work we study the k-domination number of Cartesian products of two complete graphs, which is a lower bound of the k-domination number of Cartesian product of any two graphs with the same number of vertices. In this work, we were able to find closed formula for Cartesian product of complete graphs of the same order and some values of k. We also include some results about arbitrary k. Using this finding and our result about upper bound, we were able to find bounds for arbitrary \(n,m\) .