In the present work we study the motion of microorganisms swimming by an axisymmetric distribution of surface tangential velocity in a weakly viscoelastic fluid. The second-order fluid constitutive equation is used to model the suspending fluid, while the well-known "squirmer model" [M. J. Lighthill, Comm. Pure Appl. Math. 5, 109 (1952); J. R. Blake, J. Fluid Mech. 46, 199 (1971)] is employed to describe the organism propulsion mechanism. A regular perturbation expansion up to first order in the Deborah number is performed, and the generalized reciprocity theorem from Stokes flow theory is then used, to derive analytical formulas for the squirmer velocity. Results show that "neutral" squirmers are unaffected by viscoelasticity, whereas "pullers" and "pushers" are slowed down and hastened, respectively. The power dissipated by the swimming microorganism and the "swimming efficiency" are also analytically quantified.