This work proposes novel synchronous, asynchronous, and session-based designs for energy-efficient massive multiple-input multiple-output networks to support federated learning (FL). The synchronous design relies on strict synchronization among users when executing each FL communication round, while the asynchronous design allows more flexibility for users to save energy by using lower computing frequencies. The session-based design splits the downlink and uplink phases in each FL communication round into separate sessions. In this design, we assign users such that one of the participating users in each session finishes its transmission and does not join the next session. As such, more power and degrees of freedom will be allocated to unfinished users, resulting in higher rates, lower transmission times, and hence, higher energy efficiency. In all three designs, we use zero-forcing processing for both uplink and downlink, and develop algorithms that optimize user assignment, time allocation, power, and computing frequencies to minimize the energy consumption at the base station and users, while guaranteeing a predefined maximum execution time of each FL communication round.