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In this article, we investigate the maximal bilinear Riesz means $S^{\alpha}_{*}$ associated to the sublaplacian on the Heisenberg group. We prove thatthe operator $S^{\alpha }_{*}$ is bounded from $L^{p_{1}}\times L^{p_{2}}$ into$% L^{p}$ for $2\leq p_{1}, p_{2}\leq \infty $ and $1/p=1/p_{1}+1/p_{2}$ when$% \alpha $ is large than a suitable smoothness index $\alpha (p_{1},p_{2})$.For obtaining a lower index $\alpha (p_{1},p_{2})$, we define two importantauxiliary operators and investigate their $L^{p}$ estimates,which play a keyrole in our proof.

Maximal estimates for the bilinear Riesz means on Heisenberg groups