Foundation models (FMs) are general-purpose artificial intelligence (AI) models that have recently enabled multiple brand-new generative AI applications. The rapid advances in FMs serve as an important contextual backdrop for the vision of next-generation wireless networks, where federated learning (FL) is a key enabler of distributed network intelligence. Currently, the exploration of the interplay between FMs and FL is still in its nascent stage. Naturally, FMs are capable of boosting the performance of FL, and FL could also leverage decentralized data and computing resources to assist in the training of FMs. However, the exceptionally high requirements that FMs have for computing resources, storage, and communication overhead would pose critical challenges to FL-enabled wireless networks. In this article, we explore the extent to which FMs are suitable for FL over wireless networks, including a broad overview of research challenges and opportunities. In particular, we discuss multiple new paradigms for realizing future intelligent networks that integrate FMs and FL. We also consolidate several broad research directions associated with these paradigms.