We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartley and K. Murasugi. A set of Milnor invariants for covering links is a cobordism invariant of a link, and that this invariant can distinguish some links for which the ordinary Milnor invariants coincide. Moreover, for a Brunnian link $L$, the first non-vanishing Milnor invariants of $L$ is modulo-$2$ congruent to a sum of Milnor invariants of covering links. As a consequence, a sum of linking numbers of iterated covering links gives the first non-vanishing Milnor invariant of $L$ modulo $2$.
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