This technical note is concerned with the decay rate constrained stability analysis for linear delay systems that possess cone-invariant property. In order to capture the decay rate of such systems, we introduce a nondecreasing positive function whose reciprocal represents the decay rate. Under mild assumptions on the growth rate of this function, an explicit condition is given to ensure that a cone-preserving linear system with unbounded time-varying delays is asymptotically stable with a given decay rate. As typical cases, necessary and sufficient conditions are given to characterize the decay rate when the delay is restricted by a linear, sublinear or logarithmic growth rate. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results.