It has been suggested that the current Three Lions boss is interesting those in north London.
Questions continue to be asked of Pochettino’s future, with it claimed that Argentine is starting to have his head turned amid ongoing talk of interest from Old Trafford and Real Madrid.
Were he to move on, then it could be that Southgate’s exploits in an international post bring him into contention for a return to club management.
His only previous experience at that level came during a three-year stint at Middlesbrough, in which he suffered relegation, but his stock has risen after overseeing runs to the semi-finals of the 2018 World Cup and 2019 Nations League.
Allardyce, who preceded Southgate with England before being removed from the role, told talkSPORT when quizzed on the Tottenham talk: “It depends on his own ambitions, on whether he thinks he’s got something to prove by coming to manage in the Premier League, because obviously his Middlesbrough days didn’t last very long or end too well because he got thrown in the deep end.
“Obviously he’s learnt an awful lot from then to now. I think his partnership with [assistant] Steve Holland is extremely good and that relationship is excellent.
“They may fancy a dabble back [in the Premier League], but it’s a big ask leaving England to go and manage in the Premier League.
“At some point or other, it depends on the situation of the FA being willing to let him go, Gareth wanting to go, and how much the compensation would be and whether [Tottenham chairman] Daniel Levy is willing to pay it.
“I can’t see him going before the Euros and I can’t see Mauricio going anywhere at the moment.”
Expanding on his statement regarding Pochettino, Allardyce added: “At the moment, Tottenham are a better squad than Manchester United… and that is a telling fact.
“It’s alright saying, ‘this is Manchester United, one of the biggest clubs in the world’, but has it got the team to go with it to represent Manchester United.
“It’s like Steve Bruce at the moment; he’s getting all the flak at Newcastle but is the squad good enough for Newcastle? It doesn’t look like it.”
Southgate is currently guiding England towards the 2020 European Championships, with maximum points having been taken from four qualifying outings to date.
The Three Lions will be back in action on October 11 when they take in a trip to the Czech Republic, before staying out on the road to face Bulgaria three days later.