We present a method to calculate an upper bound on the generation of entanglement in any spin system using the Fannes-Audenaert inequality for the von Neumann entropy. Our method is not only useful for efficiently estimating entanglement, but also shows that entanglement generation depends on the distance of the quantum states of the system from corresponding minimum-uncertainty spin coherent states. We illustrate our method using a quantum kicked top model, and show that our upper bound is a very good estimator for entanglement generated in both regular and chaotic regions. In a deep quantum regime, the upper bound on entanglement can be high in both regular and chaotic regions, while in the semiclassical limit, the bound is higher in chaotic regions where the quantum states diverge from the corresponding spin coherent states. Our analysis thus explains previous studies and clarifies the relationship between chaos and entanglement.