Explainability in artificial intelligence (XAI) remains a crucial aspect for fostering trust and understanding in machine learning models. Current visual explanation techniques, such as gradient-based or class-activation-based methods, often exhibit a strong dependence on specific model architectures. Conversely, perturbation-based methods, despite being model-agnostic, are computationally expensive as they require evaluating models on a large number of forward passes. In this work, we introduce Foveation-based Explanations (FovEx), a novel XAI method inspired by human vision. FovEx seamlessly integrates biologically inspired perturbations by iteratively creating foveated renderings of the image and combines them with gradient-based visual explorations to determine locations of interest efficiently. These locations are selected to maximize the performance of the model to be explained with respect to the downstream task and then combined to generate an attribution map. We provide a thorough evaluation with qualitative and quantitative assessments on established benchmarks. Our method achieves state-of-the-art performance on both transformers (on 4 out of 5 metrics) and convolutional models (on 3 out of 5 metrics), demonstrating its versatility among various architectures. Furthermore, we show the alignment between the explanation map produced by FovEx and human gaze patterns (+14\% in NSS compared to RISE, +203\% in NSS compared to GradCAM). This comparison enhances our confidence in FovEx's ability to close the interpretation gap between humans and machines.