The expected appointment of interim boss Southgate, who will meet FA bosses on Monday to discuss the job, comes at a turbulent time for the national team, with criticism of their players' behaviour during the last international break.
After playing as England defeated Scotland 3-0 in a World Cup qualifier at Wembley, captain Wayne Rooney made headlines for the wrong reasons when he was pictured, apparently intoxicated, at a wedding held at the national team's hotel the following night, three days before their friendly against Spain.
Further unconfirmed reports have claimed other players attended a local nightclub, while Liverpool duo Adam Lallana and Jordan Henderson are alleged to have driven from London to Bournemouth, where they attended a strip club, before returning to camp.
But Bevington is certain Southgate is the right man for the role after going undefeated in four matches in temporary charge, telling BBC Sport: "There is a board meeting on November 30 of the main FA board and I think Gareth will be the manager by then.
"I don't think they will be speaking to anybody else. I think Gareth is the only candidate because he has earned it, he should get it and he is the right man for the job.
"I've spent a lot of time with Gareth and his interaction with the players is fantastic. If you talk to him privately you understand what his beliefs about football are, that's why Arsene Wenger is a great admirer of Gareth, that's why his beliefs in football are first class."
Bevington is also a supporter of Southgate because he ticks the box of being an Englishman, something he feels is essential.
He added: "I found [Fabio] Capello and [Sven-Goran] Eriksson very relaxed about dealing with the media. They didn't seem to be affected by that in any way, but I genuinely believe the England team should be managed by an Englishman.
"There are number of phrases used to describe Gareth – he's a bit too soft, a safe pair of hands, a company man.
"But he is very intelligent and highly experienced football player and coach at international level. He's the most suitable person out there and the last two months, particularly the performance against Spain, have enhanced that."