In this paper we present a novel online transfer learning approach to determine the set of tests to perform, and the sequence in which they need to be performed, in order to develop an accurate diagnosis while minimizing the cost of performing the tests. Our learning approach can be incorporated as part of a clinical decision support system (CDSS) with which clinicians can interact. The approach builds on a contextual bandit framework and uses online transfer learning to overcome limitations with the availability of rich training data sets that capture different conditions, context, test results as well as outcomes. We provide confidence bounds for our recommended policies, which is essential in order to build the trust of clinicians. We evaluate the algorithm against different transfer learning approaches on real-world patient alarm datasets collected from Neurological Intensive Care Units (with reduced costs by 20%).