A lattice \(L\) of equivalence relations on a set \(A\neq \emptyset\) is called \(k\)-submodular \((k\geqq 2\) being a positive integer) if for all \(\theta, \phi, \psi \in L\) with \(\theta \subseteq \psi\) the condition \((\theta,\phi,\theta \dots)\cap \psi\subseteq \theta \vee (\phi\vee \psi)\) (where in the brackets on the left side there are \(k\) factors) is satisfied. An algebra \(\mathcal A\) is congruence \(k\)-submodular if Con \(\mathcal A\) is \(k\)-submodular. A variety \(\mathcal V\) is congruence \(k\)-submodular if each \(\mathcal A\in \mathcal V\) has this property. In the paper it is shown that the congruence \(k\)-submodularity of a variety can be characterized by Mal'tsev type conditions. Further, it is proved that a variety is congruence modular if and only if it is congruence 4-submodular. An example of a unary algebra \(\mathcal A\) is given such that \(\mathcal A\) is congruence 4-submodular, but not congruence modular.