The process (+)e- -> p (p) over bar pi(0) is studied at 20 center-of-mass energies ranging from 2.1000 to 3.0800 GeV using 636.8 pb(-1) of data collected with the BESIII detector operating at the BEPCII collider. The Born cross sections for e(+)e- -> p (p) over bar pi(0) are measured with high precision. Since the lowest center-of-mass energy, 2.1000 GeV, is less than 90 MeVabove the (p) over bar pi(0) energy threshold, we can probe the threshold behavior for this reaction. However, no anomalous threshold enhancement is found in the cross sections for e(+)e- -> p (p) over bar pi(0).