With deep neural models increasingly permeating our daily lives comes a need for transparent and comprehensible explanations of their decision-making. However, most explanation methods that have been developed so far are not intuitively understandable for lay users. In contrast, natural language explanations (NLEs) promise to enable the communication of a model's decision-making in an easily intelligible way. While current models successfully generate convincing explanations, it is an open question how well the NLEs actually represent the reasoning process of the models - a property called faithfulness. Although the development of metrics to measure faithfulness is crucial to designing more faithful models, current metrics are either not applicable to NLEs or are not designed to compare different model architectures across multiple modalities. Building on prior research on faithfulness measures and based on a detailed rationale, we address this issue by proposing three faithfulness metrics: Attribution-Similarity, NLE-Sufficiency, and NLE-Comprehensiveness. The efficacy of the metrics is evaluated on the VQA-X and e-SNLI-VE datasets of the e-ViL benchmark for vision-language NLE generation by systematically applying modifications to the performant e-UG model for which we expect changes in the measured explanation faithfulness. We show on the e-SNLI-VE dataset that the removal of redundant inputs to the explanation-generation module of e-UG successively increases the model's faithfulness on the linguistic modality as measured by Attribution-Similarity. Further, our analysis demonstrates that NLE-Sufficiency and -Comprehensiveness are not necessarily correlated to Attribution-Similarity, and we discuss how the two metrics can be utilized to gain further insights into the explanation generation process.