  Recently the CDF collaboration at the Tevatron reported a significant
discrepancy between the direct measurement of the $W$-boson mass and its
Standard Model (SM) prediction based on electroweak precision tests (EWPTs). In
this paper we explore the potential origin of this discrepancy from physics
beyond the SM. Explicitly, we work on a set of six-dimensional operators in the
SM effective field theory (SMEFT) which are relevant to the EWPTs. By fitting
to the data, we demonstrate that an upward shift in $m_W$ is driven by the
operator
$\mathcal{O}_{T}=\frac{1}{2}(H^{\dagger}\overset{\text{$\leftrightarrow$}}{D}_{\mu}H)^2$
with a coefficient $c_T ({\rm TeV}/\Lambda)^2 \gtrsim 0.01$. This suggests that
the new physics scale favored by the CDF data should be multiple TeV for
tree-level effects and sub TeV for loop-level effects. One simple example is to
introduce a hypercharge-free electroweak triplet scalar which can raise the
$c_T$ value at tree level. We also study the potential to further test the
relevant SMEFT by measuring Higgs-coupling, $m_W$ and other EWPTs at future
circular $e^-e^+$ colliders.
