The results of a search for the rare two-body charmless baryonic decays B0 → pp̄ and B0s → pp̄ are reported. The analysis uses a data sample, corresponding to an integrated luminosity of 0.9 fb-1, of pp collision data collected by the LHCb experiment at a centre-of-mass energy of 7 TeV. An excess of B0 → pp̄ candidates with respect to background expectations is seen with a statistical significance of 3.3 standard deviations. This is the first evidence for a two-body charmless baryonic B0 decay. No significant B0s → pp̄ signal is observed, leading to an improvement of three orders of magnitude over previous bounds. If the excess events are interpreted as signal, the 68.3% confidence level intervals on the branching fractions are (Equation) where the first uncertainty is statistical and the second is systematic.