Evolutionary games describe the repeated random interactions of individuals with others in a large population. In these interactions, individuals have a choice between strategies that enable them to pursue a group interest (cooperate) or pursue a self-interest (defect). When the individuals have a capacity to change this selection of strategy based on their interactions, and the decisions are based on performance that is influenced by the choice of others, then the evolution has been shown to lead to a stable percentage of cooperators over time. The dynamics of such an evolution of strategies can be described by individuals adjusting their strategy either (i) based on comparison of their performance to that of others (imitation update rules) or (ii) based on assessing their performance against a goal (aspiration update rules). In either case, these dynamics are typically modeled as an instantaneous reaction to the observations made by the individuals. However, in real physical and social systems, there are often non-negligible time delays between the observations and the ability of an individual to adapt. When these time delays become large enough, they can create an instability in the dynamics of the system strategies pursued by the individuals (as measured by the percentage of individuals following the cooperation strategy). We investigate the initiation of this instability via a Hopf bifurcation in the time delay parameter. In addition to explaining the transition, this analysis also yields guidance on how much time delay is tolerable in order to guarantee convergence of the group behavior. Both analytical and numerical results are presented to illustrate the phenomenon.