We investigate the detectability of soliton cores predicted by self-interacting scalar field dark matter (SI-SFDM) in the Thomas--Fermi regime using strong gravitational lensing by galaxy clusters. We construct semi-analytical density profiles composed of a solitonic core and an outer NFW halo, and compute deflection angles and excess surface mass density for clusters with $M_{200}=2\times10^{15}M_\odot$ and $2\times10^{14}M_\odot$. Massive solitons generate deflection deviations larger than $2''$ at radii of 10--30 kpc, within the precision of current strong-lensing observations. Comparisons with A2390 and DES weak-lensing measurements show that SI-SFDM remains consistent with large-scale mass profiles while central densities constrain the soliton mass to $M_{\rm sol}\lesssim10^{12}M_\odot$. Our results indicate that strong lensing provides a promising probe of scalar-field interactions in cluster cores.