The radial dynamics of a spherical bubble in a compressible viscoelastic liquid is studied by means of a simplified singular-perturbation method to first-order in the bubble-wall Mach number. The three-parameter linear Oldroyd model is adopted to describe the viscoelastic properties of the liquid. The equation of motion for the bubble radius and the pressure equation are derived and numerical calculations are conducted for the case of bubble collapse in a constant-pressure field. It is concluded that the rheology of the liquid strongly influence the behaviour of bubbles only for values of the Reynolds number (Re = R(0)root p(infinity)rho(infinity)/eta where R-0 is the initial radius, p(infinity) is the undisturbed liquid pressure, rho(infinity) is the liquid density and eta is the liquid viscosity) smaller than 10(2) while for Re greater than or equal to 10(2), sound emission is the main damping mechanism in spherical bubble dynamics. In both cases, the 1/r law of pressure attenuation through the liquid is not affected by the viscoelastic properties of the liquid. The effect of polymer additives on spherical bubble collapse is also discussed.