The Sun says the Boro boss has offered Everton skipper Neville, 32, the chance to join his backroom staff.
Southgate, 38, believes his one-time international team-mate would be perfect for a player-coaching role, and hopes the chance of cutting his teeth while still at a good level will tempt him.
Yet it seems certain the former England centre-back's efforts will fall on deaf ears, with Neville committed to Everton.
Neville has never made any secret of the fact he plans to go into coaching or management when he finally hangs up his boots.
But he only signed a new four-year contract 12 months ago, insisting at the time that he wanted to finish his playing days at Goodison.