In this article, we construct the color singlet-singlet-singlet interpolating current with $I\left(J^P\right)=\left(3/2\right)\left(1^{-}\right)$ to study the $D{\overline{D}}^{⁎}K$ system through QCD sum rules approach. In calculations, we consider the contributions of the vacuum condensates up to dimension-16 and employ the formula $\mu =\sqrt{M_{X/Y/Z}^2-{\left(2M_c\right)}^2}$ to choose the optimal energy scale of the QCD spectral density. The numerical result $M_Z=4.71_{-0.11}^{+0.19}GeV$ indicates that there exists a resonance state $Z$ lying above the $D{\overline{D}}^{⁎}K$ threshold to saturate the QCD sum rules. This resonance state $Z$ may be found by focusing on the channel $J/\psi \pi K$ of the decay $B\to J/\psi \pi \pi K$ in the future.