That's according to Ray Parlour, but he adds that Southgate should be judged on his performance at the tournament in Qatar before he is sacked.
The national team boss has taken England to a World Cup semi-final and European Championship final in successive major tournaments.
But supporters and pundits have slated his team's performances in the Nations League, where two defeats to Hungary and draws with Germany and Italy have left the Three Lions sitting rock bottom of their group.
And Parlour believes Southgate will be gone if this wretched form continues into the World Cup later this year and performances in what will be a physically demanding tournament in Qatar aren't up to scratch.
He told the talkSPORT Breakfast: "It's not ideal to lose 4-0 home, is it? That's not a great result.
"It'll be a situation where I'll judge Gareth on the World Cup, I think you've got to give him a chance.
"But if he fails at the World Cup, then obviously his jobs up. That's probably it."
The 4-0 defeat to Hungary left some supporters calling for Southgate to be sacked before the World Cup, with many complaining about his negative style of football.
However, when asked about who could replace him, Parlour struggled for names, with Gerrard and Potter the two he threw into the ring.
Potter has done a fantastic job at Brighton, playing attractive football on the south coast, leading them to a ninth-place Premier League finish last season.
As for Gerrard, after winning the Scottish Premiership title with Rangers, the ex-England captain is beginning to find his feet at Aston Villa and is continuing to develop as a head coach.
And, after Mauricio Pochettino and Brendan Rodgers were both suggested, Parlour stated that the next appointment will preferably be English - but he also wants the best man for the job.
When asked about who could take the job if he leaves, Parlour added: "In the future, someone like Steven Gerrard.
"Right now, you'd probably look at someone like Graham Potter, wouldn't you?
"Ideally, you want an Englishman to manage England but you also want the best coach."