Bubble nucleation and growth are the key steps in polymer foam processes. The thermodynamic specification of the critical cluster size for nucleation has been commonly used as the starting point for bubble growth analyses. To initiate the solution, a perturbation of the critical radius is required and the selection of this perturbation is somewhat arbitrary. Depending on the size and nature of the perturbation, significant differences can arise in the predicted bubble growth dynamics. In the present study, we consider a cluster to be nucleated when it reaches a supercritical state where the thermodynamic fluctuations cannot drive it back to subcritical conditions. For this purpose, a critical region is defined within which the thermodynamic fluctuations have significant effects on cluster growth. The state of the cluster at the upper bound of the critical region then provides the initial conditions for the bubble growth analysis. A perturbation solution in terms of the Peclet number has been used to determine the state of the cluster at the upper bound of the critical region (at nucleation) as well as the initial bubble growth stage. The predictions of the analytical solution compare well with the predictions from numerical solutions.