Bayesian optimality criteria provide a robust design strategy to parameter misspecification. We develop an approximate design theory for Bayesian D‐optimality for nonlinear regression models with covariates subject to measurement errors. Both maximum likelihood and least squares estimation are studied, and explicit characterisations of the Bayesian D‐optimal saturated designs for the Michaelis–Menten, Emax and exponential regression models are provided. Several data examples are considered for the case of no preference for specific parameter values, where Bayesian D‐optimal saturated designs are calculated using the uniform prior and compared with several other designs, including the corresponding locally D‐optimal designs, which are often used in practice. Copyright © 2017 John Wiley & Sons, Ltd.