We report the results of a search for axions from the 14.4 keV M1 transition from ⁵⁷Fe in the core of the sun using the axio-electric effect in TeO₂ bolometers. The detectors are 5 × 5 × 5 cm³ crystals operated at about 10 mK in a facility used to test bolometers for the CUORE experiment at the Laboratori Nazionali del Gran Sasso in Italy. An analysis of 43.65 kg⋅d of data was made using a newly developed low energy trigger which was optimized to reduce the energy threshold of the detector. An upper limit of 0.58 c⋅kg⁻¹⋅d⁻¹ is established at 95% C.L., which translates into lower bounds f_A ≥ 3.12 × 10⁵ GeV 95% C.L. (DFSZ model) and f_A ≥ 2.41 × 10⁴ 95% C.L. (KSVZ model) on the Peccei-Quinn symmetry-breaking scale, for a value of S = 0.5 of the flavor-singlet axial vector matrix element. These bounds can be expressed in terms of axion masses as m_A ≤ 19.2 eV and m_A ≤ 250 eV at 95% C.L. in the DFSZ and KSVZ models respectively. Bounds are given also for the interval 0.35 ≤ S ≤ 0.55.