Accurate characterization of heat transfer in a wellbore during drilling, which includes fluid circulation, is important for wellbore stability analysis. In this work, a pseudo-3D model is developed to simultaneously calculate the heat exchange between the flowing fluid and the surrounding media (drill pipe and rock formation) and the in-plane thermoelastic stresses. The cold drilling fluid descends through the drill pipe at constant injection rates and returns to the ground surface via the annulus. The fluid circulation will decrease the wellbore bottom temperature and reduce the near-wellbore high compressive stress, potentially leading to tensile fracturing of the well. The governing equations for the coupled heat transfer stress problem are formulated to ensure that the most important parameters are taken into account. The wellbore is subject to a non-hydrostatic in situ far-field stress field. In modeling heat exchange between fluid and surrounding media, the heat transfer coefficients are dependent on fluid properties and flow behavior. Analytical solutions in the Laplace space are obtained for the temperatures of the fluid in both the drill pipe and annulus and for the temperature and stress changes in the formation. The numerical results in the time domain are obtained by using an efficient inversion approach. In particular, the near-well stresses are compared for the cases with fixed and time-dependent cooling wellbore conditions. This comparison indicates that the using a fixed temperature wellbore conditions may over-estimate or under-estimate the bottom-hole stress change, potentially leading to wellbore stability problems.