The Eulerian extension number of any graph~\(H\) (i.e. the minimum number of edges needed to be added to make~\(H\) Eulerian) is at least~\(t(H),\) half the number of odd degree vertices of~\(H.\) In this paper we consider an inhomogenous random graph~\(G\) whose edge probabilities need not all be the same and use an iterative probabilistic method to obtain sufficient conditions for the Eulerian extension number of~\(G\) to grow \emph{linearly} with~\(t(G).\) We derive our conditions in terms of the average edge probabilities and edge density and also briefly illustrate our result with an example.