Circular electron positron colliders, such as the CEPC and FCC-ee, have been proposed to measure Higgs boson properties precisely, test the Standard Model, search for physics beyond the Standard Model, and so on. One of the important goals of these colliders is to measure the $W$ boson mass with great precision by taking data around the $W$-pair production threshold. In this paper, the data-taking scheme is investigated to maximize the achievable precisions of the $W$ boson mass and width with a threshold scan, when various systematic uncertainties are taken into account. The study shows that an optimal and realistic data-taking scheme is to collect data at three center-of-mass energies and that precisions of 1.0 MeV and 3.4 MeV can be achieved for the mass and width of the $W$ boson, respectively, with a total integrated luminosity of $mathcal{L}=3.2$~mbox{ab}$^{-1}$ and several assumptions of the systematic uncertainty sources.
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