But 46-year-old Southgate says he can turn things around.
The Telegraph also revealed that Allardyce negotiated in principle a deal to travel to the Far East as a paid ambassador for the fictitious businessmen.
A third agent, former Manchester United player Scott McGarvey, said he gave an associate £30,000 to secure a player transfer.
"I also know how proud he was he had got it and I am gutted for him that he has lost his job through events off the pitch, rather than on the pitch". On reflection it was a silly thing to do. "Entrapment has won on this occasion".
Allardyce added he had simply been helping out a longstanding associate.
"Unfortunately it was an error in judgment on my behalf. I apologize to those and all concerned in this unfortunate situation I've put myself in".
Allardyce's departure was essential to preserve the Football Association's integrity as "guardians of the game", the governing body's senior figures said on Tuesday.
Klinsmann had been linked to the position before Allardyce was hired but was never contacted by the FA, despite rumors to the contrary following Roy Hodgson's resignation following Euro 2016.
The Daily Mail claim he was subjected to an "intense interrogation" by FA chairman Greg Dyke and chief executive Martin Glenn - but the paper claims Glenn and FA technical director Dan Ashworth were reluctant to lose the manager they appointed back in July.
"Now the best man available isn't always the best man for the job and I think they did make a mistake, they did make an error of judgement". I have cooperated fully in this regard.
Following Sam Allardyce's exit, Southgate will be in-charge of the team for its next four games which include three 2018 Fifa World Cup qualifier fixtures.
'It's sad but I have zero respect for anyone caught because it is people stealing money, ' said Coleman.
"Where they go from here, I do not know".
Asked if that was his last job in football he said: "Who knows". We've seen it in governments and all other sports, whether it's drug-taking in athletics, cycling or gymnastics.
Clarke stressed there will be no rush to appoint a permanent successor to Allardyce, with Gareth Southgate having been given the job on an interim basis for the next four games. The agreement was done very amicably with the FA. I'm going on holiday to reflect.
"They have shown me great faith and I have a good team I have a good feeling about".
Allardyce, or "Big Sam" as he is known throughout the media, spent the 1984/85 season at Huddersfield Town as a player, helping Town overcome a crop of injuries to finish 13th in what was the old Division Two.