A parameter study was carried out on the design variables of a flexure bearing to be used in a Stirling engine with a fixed axial displacement and a fixed outer diameter. A design method was developed in order to assist identification of the optimum bearing configuration. This was achieved through a parameter study of the bearing carried out with ANSYS®. The parameters varied were the number and the width of the arms, the thickness of the bearing, the eccentricity, the size of the starting and ending holes, and the turn angle of the spiral. Comparison was made between the different designs in terms of axial and radial stiffness, the natural frequency, and the maximum induced stresses. Moreover, the Finite Element Analysis (FEA) was compared to theoretical results for a given design. The results led to a graphical design method which assists the selection of flexure bearing geometrical parameters based on pre-determined geometric and material constraints.