In scientific and engineering applications, often sufficient low-cost low-fidelity data is available while only a small fractional of high-fidelity data is accessible. The multi-fidelity model integrates a large set of low-cost but biased low-fidelity datasets with a small set of precise but high-cost high-fidelity data to make an accurate inference of quantities of interest. Under many circumstances, the number of model input dimensions is often high in real applications. To simplify the model, dimension reduction is often used. The gradient-free active subspace is employed in this research for dimension reduction. In this work, a novel predictive model for high-dimensional nonlinear problems by integrating the nonlinear multi-fidelity Gaussian progress regression and the gradient-free active subspace method is put forward. Numerical results demonstrated that the proposed approach can not only perform effective dimension reduction on the original data but also obtain accurate prediction results thanks to the effective dimension reduction procedure.