Direct-contact heat transfer involves the exchange of heat between two fluids by bringing them into intimate contact with each other. While one of the fluids involved is the continuous phase or the quiescent phase, the other, which is intimately mixed with the former, is called as the dispersed phase. The advantage of direct-contact heat transfer is that it generates a very high value of overall heat transfer coefficient resulting in higher heat transfer efficiency. This overall heat transfer coefficient is at its maximum at the instant when the two fluids (phases) make their contact. The value of the heat transfer coefficient decreases rapidly due to the additional heat transfer resistances built up such as the condensate layer and so on. This rapid process has made experimental measurement of the initial heat transfer coefficient illusive. So far there has been only one such analytical expression that has been derived to describe this initial process. The same authors also devised an experimental technique for measuring the direct contact heat transfer coefficient during this initial phase. However, while the experimental technique of the abovementioned authors was adequate, their analytical expression to evaluate the direct-contact heat transfer coefficient was incorrect. In this paper, five mathematical expressions have been presented for estimation of the direct-contact heat transfer coefficient for a condensing vapour bubble. Of these, three have been exclusively derived here and the other two are modifications from the already existing expressions. Comparisons have been made with the available experimental results and the physics revealed by the comparisons are described.