In this article, we investigate the maximal bilinear Riesz means $S_{\ast}^{\alpha}$ associated to the sublaplacian on the Heisenberg group. We prove that the operator $S_{\ast}^{\alpha}$ 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 important auxiliary operators and investigate their $L^{p}$ estimates, which play a key role in our proof.