Southgate's contract runs until the end of EURO 2024 but he is considering his future after England were knocked out by France in the quarterfinals of the Qatar 2022 FIFA World Cup.
Southgate has been in charge of England since 2016 and took the Three Lions to the semi-finals of the 2018 World Cup and final of EURO 2020, losing to Italy.
"There will be those who are listening, who say, 'You've got to have someone who gets us over the line, you've got to have someone who has a winning mentality, you've got to change the manager'," Neville told Sky Sports.
"I get that, it sounds beautiful and wonderful, but I remember us getting rid of Bobby Robson in 1990. I remember us getting rid of Terry Venables in 1996.
"It didn't go particularly well straight after those tournaments and I think Gareth is doing a very good job and one more tournament for me feels right.
"Hopefully, this team can stick together and evolve. I think it evolved in this tournament. We were more progressive, more positive in matches, and we go for it again.
"He has seen it all, and I think he should be kept within the system to design the future. He has been there for 10 years. I would like him to stay with the FA beyond his coaching role."
