<p>This work introduces a numerical algorithm based on the spectral Tau method for solving the time-fractional bioheat transfer equation, which describes how heat moves through biological tissues over time. A specific set of Horadam polynomials is employed as trial functions. Some explicit formulas for the Horadam polynomials used, such as their explicit expressions for the integer and fractional-order derivatives and some specific definite integrals, are used in designing the numerical algorithm for solving the fractional bioheat equation. Comprehensive convergence and error analyses are conducted to validate the accuracy of the proposed Horadam expansion. Several illustrative examples are provided to ensure the applicability and accuracy of the proposed method.</p>