Quantitative diagnostics on the thermal states of biological bodies and optimal dosimetry in tumor hyperthermia require general solutions for the bioheat equation. An algorithm based on the dual reciprocity boundary element method (DRBEM) is developed to solve the integral inverse or direct bioheat transfer problems. Using this algorithm, thermal states of the biological bodies, reflecting physiological conditions, can be correlated to the temperature or heat flux detected at the skin surface. The extension of the algorithm to the non-linear bioheat model and boundary conditions is also discussed. Various cases are studied to illustrate its applications. This algorithm can be used as an effective tool for thermal diagnostics in medical practices.