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In this paper we present a calculation of the sensitivity of the CUOREdetector to the monoenergetic $14.4$ keV solar axions emitted by the M1 nucleartransition of$~^{57}$Fe in the Sun and detected by inverse coherentBragg-Primakoff conversion in single-crystal $TeO_2$ bolometers. The expectedcounting rate is calculated using density functional theory for the electroncharge density of $TeO_2$ and realistic background and energy resolution ofCUORE. Monte Carlo simulations for $5$ y $\times$ $741$ kg=$3705-$kg$\cdot$y ofexposure are analyzed using time correlation of individual events with thetheoretical time-dependent counting rate. We find an expected model-independentlimit on the product of the axion-photon coupling and the axion-nucleoncoupling $g_{a\gamma\gamma}g_{aN}^{\text{eff}}<1.105\times 10^{-16}$ /GeV foraxion masses less than 500 eV with $95\%$ confidence level.

Sensitivity of the CUORE detector to $14.4$ keV solar axions emitted by the M1 nuclear transition of$~^{57}$Fe