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