Monotone systems comprise an important class of dynamical systems that are of interest both for their wide applicability and because of their interesting mathematical properties. It is known that under the property of quasimono-tonicity time-delayed systems become monotone, and some remarkable properties have been reported for such systems. These include, for example, the fact that for linear systems global asymptotic stability of the undelayed system implies global asymptotic stability for the delayed system under arbitrary bounded delays. Nevertheless, extensions to nonlinear systems have thus far relied on various restrictive conditions, such as homogeneity and subhomogeneity, and it has been conjectured that these can be relaxed. Our aim in this paper is to show that this is feasible for a general class of nonlinear monotone systems, by deriving asymptotic stability results in which simple properties of the undelayed system lead to delay-independent stability. In particular, one of our results is to show that if the undelayed system has a convergent trajectory that is unbounded in all components as t → -∞ then the system is globally asymptotically stable for arbitrary time-varying delays. This follows from a more general result derived in the paper where delay-independent regions of attraction are quantified from the asymptotic behavior of individual trajectories of the undelayed system. This result recovers various known delay-independent stability results, and several examples are included in the paper to illustrate the significance of the proposed stability conditions.