  We perform cosmological zoom-in simulations of $19$ relaxed cluster-mass
haloes with the inclusion of adiabatic gas in the cold dark matter (CDM) and
self-interacting dark matter (SIDM) models. These clusters are selected as
dynamically relaxed clusters from a parent simulation with $M_{\rm 200} \simeq
1\operatorname{-}3\times 10^{15}\,{\rm M}_{\odot}$. Both the dark matter and
the intracluster gas distributions in SIDM appear more spherical than their CDM
counterparts. Mock X-ray images are generated based on the simulations and are
compared to the real X-ray images of $84$ relaxed clusters selected from the
Chandra and ROSAT archives. We perform ellipse fitting for the isophotes of
mock and real X-ray images and obtain the ellipticities at cluster-centric
radii of $r\simeq 0.1\operatorname{-}0.2\,R_{\rm 200}$. The X-ray isophotes in
SIDM models with increasing cross-sections are rounder than their CDM
counterparts, which manifests as a systematic shift in the distribution
function of ellipticities. Unexpectedly, the X-ray morphology of the observed
non-cool-core clusters agrees better with SIDM models with cross-section
$(\sigma/m)= 0.5\operatorname{-}1~{\rm cm}^2/{\rm g}$ than CDM and SIDM with
$(\sigma/m)=0.1\,{\rm cm}^2/{\rm g}$. Our statistical analysis indicates that
the latter two models are disfavored at the $68\%$ confidence level (as
conservative estimates). This conclusion is not altered by shifting the radial
range of measurements or applying temperature selection criterion. However, the
primary uncertainty originates from the lack of baryonic physics in the
adiabatic model, such as cooling, star formation and feedback effects, which
still have the potential to reconcile CDM simulations with observations.
