Model-based clustering represents one of the fundamental procedures in a statistician's toolbox. Within the model-based clustering framework, we consider the case where the kernel distribution of nonparametric mixture models is available only up to an intractable normalizing constant, in which most of the commonly used Markov chain Monte Carlo methods fail to provide posterior inference. To overcome this problem, we propose an approximate Bayesian computational strategy, whereby we approximate the posterior to avoid the intractability of the kernel. By exploiting the structure of the nonparametric prior, our proposal combines the use of predictive distributions as a proposal with transport maps to obtain an efficient and flexible sampling strategy. Further, we illustrate how the specification of our proposal can be relaxed by introducing an adaptive scheme on the degree of approximation of the posterior distribution. Empirical evidence from simulation studies shows that our proposal outperforms its main competitors in terms of computational times while preserving comparable accuracy of the estimates.