Using 2.25×108J/ψ events collected with the BESIII detector at the BEPCII storage rings, we observe for the first time the process J/ψ→p¯pa0(980), a0(980)→π0η with a significance of 6.5σ (3.2σ including systematic uncertainties). The product branching fraction of J/ψ→p¯pa0(980)→p¯pπ0η is measured to be (6.8±1.2±1.3)×10−5, where the first error is statistical and the second is systematic. This measurement provides information on the a0 production near threshold coupling to p¯p and improves the understanding of the dynamics of J/ψ decays to four-body processes.