A theoretical investigation has been conducted for the prediction of the critical height at the onset of gas entrainment during single discharge from a stratified, two-phase region through a side branch with a finite diameter. Two different models have been developed, a simplified point-sink model and a three-dimensional finite-branch model. The two models are based on a new criterion for the onset of gas entrainment. The results of the predicted critical heights at the onset of gas entrainment showed that the finite-branch model approaches the physical limits at low Froude numbers. However, as the values of the Froude number increased, the predictions of both models eventually converged to the same value. Based on the results of the models, the critical height corresponding to the onset of gas entrainment was found to be a function of Froude number and fluid densities. The results of both models are compared with available experimental data. The comparisons illustrate a very good agreement between the measured and predicted values.