Using a sample of 1.31×109 J/ψ events collected with the BESIII detector at the BEPCII collider, the decays J/ψ→ϕπ+π−π0 and J/ψ→ϕπ0π0π0 are investigated. The isospin-violating decay J/ψ→ϕπ0f0(980) with f0(980)→ππ is observed for the first time. The width of the f0(980) obtained from the dipion mass spectrum is found to be much smaller than the world average value. In the π0f0(980) mass spectrum, there is evidence of f1(1285) production. By studying the decay J/ψ→ϕη′, the branching fractions of η′→π+π−π0 and η′→π0π0π0, as well as their ratio, are also measured.