Bounded confidence opinion dynamics are dynamic networks in which agents are connected if their opinions are similar, and each agent updates her opinion as the average of the neighbors’ opinions. In homogeneous asymmetric Heglselmann–Krause (HK) models, all agents have the same confidence thresholds which could be different for the selection of upper and lower neighbors. This paper provides conditions for the convergence of the opinions to consensus and to clustering for this class of HK models. A new tighter bound on the time interval for reaching the steady state is also provided.