The 47-year-old has two years remaining on his current deal, but the FA wants to reward his success in leading England to the semi-finals of the 2018 World Cup.
FA chief executive Martin Glenn is aware that the governing body cannot compete with the financial lure of Premier League management but believes that Southgate shares his sense of "unfinished business" with the national team.
"We'd like him to stay beyond 2020 and I think he would like that too but we haven't spoken about it in any depth because that would be a contract discussion and he's on holiday now so we'll talk about that when he gets back," Glenn is quoted by The Guardian as saying.
"It's our job to make it competitive to make sure him and all of our people are competitively rewarded. We can never compete with a Premier League club in terms of pay.
"He's really blossomed. It was fantastic seeing him first hand in pressure situations. He fits the definition of the modern manager we want. He gets that it's not just picking the team but all the aspects behind it and has benefited from a world-class support team. I'm sure he'd be the first to say that.
"The World Cup makes our work feel justified. It's reinforced a belief we're doing a lot right. Our target is to be a top-four team in the world and we're not there yet, so there's unfinished business."