We report the first observation of $a^{0}_{0}(980)$-$f_{0}(980)$ mixing in the decays of $J/psitophi f_{0}(980)tophi a^{0}_{0}(980)tophietapi^{0}$ and $chi_{c1}to a^{0}_{0}(980)pi^{0}to f_{0}(980)pi^{0}topi^{+}pi^{-}pi^{0}$, using data samples of $1.31times10^{9}$ $J/psi$ events and $4.48times10^{8}$ $psi(3686)$ events accumulated with the BESIII detector. The signals of $f_{0}(980)to a^{0}_{0}(980)$ and $a^{0}_{0}(980)to f_{0}(980)$ mixing are observed at levels of statistical significance of $7.4sigma$ and $5.5sigma$, respectively. The corresponding branching fractions and mixing intensities are measured and the constraint regions on the coupling constants, $g_{a_{0}K^{+}K^{-}}$ and $g_{f_{0}K^{+}K^{-}}$, are estimated. The results improve the understanding of the nature of $a^{0}_{0}(980)$ and $f_{0}(980)$.
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