  We explore the extent to which future precision measurements of the Standard
Model (SM) observables at the proposed $Z$-factories and Higgs factories may
have impacts on new physics beyond the Standard Model, as illustrated by
studying the Type-I Two-Higgs-doublet model (Type-I 2HDM). We include the
contributions from the heavy Higgs bosons at the tree-level and at the one-loop
level in a full model-parameter space. While only small $\tan\beta$ region is
strongly constrained at tree level, the large $\tan\beta$ region gets
constrained at loop level due to $\tan\beta$ enhanced tri-Higgs couplings. We
perform a multiple variable global fit with non-alignment and non-degenerate
masses. We find that the allowed parameter ranges could be tightly constrained
by the future Higgs precision measurements, especially for small and large
values of $\tan\beta$. Indirect limits on the masses of heavy Higgs bosons can
be obtained, which can be complementary to the direct searches of the heavy
Higgs bosons at hadron colliders. We also find that the expected accuracies at
the $Z$-pole and at a Higgs factory are quite complementary in constraining
mass splittings of heavy Higgs bosons. The typical results are
$|\cos(\beta-\alpha)| < 0.05, |\Delta m_\Phi | < 200\ {\rm GeV}$, and
$\tan\beta \gtrsim 0.3$. The reaches from CEPC, FCCee and ILC are also
compared, for both Higgs and $Z$-pole precision measurements. Comparing to the
Type-II 2HDM, the 95\% C.L. allowed range of $\cos(\beta-\alpha)$ is larger,
especially for large values of $\tan\beta$.
