Summary: This paper is concerned with the density-suppressed motility model: \[ u_t=\Delta \left( \displaystyle \frac{u^m}{v^{\alpha}}\right) +\beta uf(w),\quad v_t=D\Delta v-v+u,\quad w_t=\Delta w-uf(w) \] in a smoothly bounded convex domain \(\Omega \subset\mathbb{R}^2\), where \(m>1\), \(\alpha>0\), \(\beta >0\) and \(D>0\) are parameters, the response function \(f\) satisfies \(f\in C^1([0,\infty)), f(0)=0, f(w)>0\) in \((0,\infty)\). This system describes the density-suppressed motility of Eeshcrichia coli cells in the process of spatio-temporal pattern formation via so-called self-trapping mechanisms. Based on the duality argument, it is shown that for suitable large \(D\) the problem admits at least one global weak solution \((u, v, w)\) which will asymptotically converge to the spatially uniform equilibrium \((\overline{u_0}+\beta \overline{w_0},\overline{u_0}+\beta \overline{w_0},0)\) with \(\overline{u_0}=\frac{1}{|\Omega |}\int_{\Omega}u(x,0)dx\) and \(\overline{w_0}=\frac{1}{|\Omega |}\int_{\Omega}w(x,0)dx\) in \(L^\infty (\Omega )\).