The work presented here explores the application of computational argumentation and argumentation-based dialogue to the domain of clinical consultation, with particular focus on self-management of chronic health conditions. Chronic conditions are characterised by regular and sometimes frequent monitoring of various biometric signs, along with a prescribed regimen of diet, exercise and medication. A variety of different types of exchanges can occur between a patient and their health care provider(s) when managing chronic conditions. In this paper, we propose the application of existing work on agent-based dialogue systems to support wellness consultation for a patient, as a supplement to typical periodic interactions with health care professionals. The system is designed to provide decision support and explanation for recommended actions. Explanations are supported by argumentation-based dialogues using a novel combination of argumentation schemes and explanation templates. The approach is illustrated through the use of a clinical scenario.