In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation}

for $x\in \mathbb{H}^n$, where $0<\alpha<Q=2n+2$, $n\geq3$, $\beta\geq0$, $\alpha+\beta<Q$ and p,q,r > 1 satisfying $\frac{1}{p+1} + \frac{1}{q+1} + \frac{1}{r+1} = \frac{Q+α+β}{Q}$. We show that positive solution triples (u,v,w)∈L^{p+1}($\mathbb{H}^{n}$)×
L^{q+1}($\mathbb{H}^{n}$)×L^{r+1}($\mathbb{H}^{n}$) are bounded and they converge to zero when |x|→∞.