It is recently found that viscoelastic pipe flows can be linearly unstable, leading to the possibility of a supercritical transition route, in contrast to Newtonian pipe flows. Such an instability is referred to as the centre mode, which was studied numerically by Chaudhary et al. (J. Fluid Mech., vol. 908, 2021, p. A11) based on an Oldroyd-B model. In this paper, we are interested in expanding the parameter space investigated and exploring the asymptotic scalings related to this centre instability in the Oldroyd-B viscoelastic pipe flow. It is found from the asymptotic analysis that the centre mode exhibits a three-layered asymptotic structure in the radial direction, a wall layer, a main layer and a central layer, which are driven by pure viscosity, axial and/or radial pressure gradient, and a combined effect of viscosity and elasticity, respectively. Depending on the relations of the control parameters, two regimes, the long-wavelength and short-wavelength centre instabilities, emerge, for which the central-layer thicknesses are of different orders of magnitude. Our large-Reynolds-number asymptotic predictions are compared to the numerical solutions of the original eigenvalue system, and favourable agreement is achieved, especially when the parameters approach their individual limits. In addition to revealing the dominant factors and their balances, the asymptotic analysis describes the instability system by reducing the number of control parameters, and furthermore explaining the collapse of the numerical results for different re-scalings.