An edge $e$ in a matching covered graph $G$ is removable if $G-e$ is matching covered, which was introduced by Lovász and Plummer in connection with ear decompositions of matching covered graphs. A brick is a non-bipartite matching covered graph without non-trivial tight cuts. The importance of bricks stems from the fact that they are building blocks of matching covered graphs. Carvalho et al. [Ear decompositions of matching covered graphs, Combinatorica, 19(2):151-174, 1999] showed that each

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... ick other than $K_4$ and $\overline{C_6}$ has $Δ-2$ removable edges, where $Δ$ is the maximum degree of $G$. We show that every cubic brick $G$ other than $K_4$, $\overline{C_6}$ and a particular graph of order 8 has a matching of size at least $|V(G)|/7$, each edge of which is removable in $G$. Moreover, if $G$ is a 3-edge-connected near-bipartite cubic graph, then $G$ has a matching of size at least $|V(G)|/2-3$, each edge of which is removable in $G$. The lower bounds for the two sizes of matchings are sharp.