The Florida-based team are due to make their MLS bow next season, with the former England and Manchester United winger eyeing a few big names as he searches for Miami's first manager.
Moyes - who played for Celtic between 1980 and 1983 - has had management spells with Preston, Everton, Manchester United, Real Sociedad, Sunderland and West Ham, but has been out of work since leaving the Hammers in 2018 after a poor run of form.
The Scot was linked with the Celtic job following Brendan Rodgers' departure and was touted for the Scotland job after Alex McLeish was relieved of his duties.
Former Ibrox midfielder Gattuso, who has had spells managing Palermo, OFI Crete, Pisa and AC Milan, is also understood to be under consideration for the role.
Ex-Belgium No.2 Thierry Henry is also rumoured to be in the running along with Carlo Ancelotti, but Inter Miami chiefs could have a task on their hands trying to pry the latter away from Napoli.