  Following the observation of the fully-heavy tetraquark candidates $X(6900)$
and possible $X(6300)$ in the di-$J/\psi$ spectrum by the LHCb Collaboration,
we investigate the near-threshold dynamics of the pseudoscalar quarkonium pairs
in a coupled-channel approach in the di-$\eta_c$ channel. We show that the
Pomeron exchange mechanism should be a general dynamics in the near-threshold
heavy quarkonium pair interactions. In the di-$\eta_c$ channel, a
coupled-channel system involving the di-$\eta_c$, $\eta_c$-$\eta_c(2S)$ and
di-$\eta_c(2S)$ interactions can be established. Their $S$-wave interactions
near threshold via the Pomeron exchange potential can produce near-threshold
resonance poles similar to the di-$J/\psi$, $J/\psi$-$\psi(2S)$ and
di-$\psi(2S)$ coupled channels. Interestingly, we find here that the pole
positions are more shifted to the corresponding thresholds and turn out to be
different from those seen in the di-$J/\psi$ spectrum. Taking into account the
suppression of the heavy quark spin flips in the transitions between two vector
and two pseudoscalar heavy quarkonium systems, the enhancements in the
di-$\eta_c$ spectrum should be different from those seen in the di-$J/\psi$
channel. Experimental study of the di-$\eta_c$ channel should be useful for
disentangling the nature of the fully-heavy tetraquark systems.
