We consider the Cauchy problem for the nonlinear heat equation
                                                                       $u_t=\Delta u+f(u), \quad x\in \mathbb R^N,\ t>0,$
where $N\ge 2$ and $f$ is a $C^1$ function satisfying minor nondegeneracy conditions. Our goal is to describe the large-time behavior of bounded solutions whose initial data are radially symmetric and have a finite limit $\zeta$ as $|x|\to\infty$. In the present paper, we examine the following two cases: $f(\zeta)\ne 0$, or $f(\zeta)=0$ and $\zeta$ is a stable equilibrium of the equation $\dot \xi=f(\xi)$. We prove that bounded solutions with such initial data are quasiconvergent: as $t\to \infty$, they approach a set of steady states in the topology of $L^\infty_{loc}(\mathbb R^N)$.