The 61-year-old held discussions with the FA over the England manager's job last week, which became vacant when Roy Hodgson stepped down after a disappointing Euro 2016 finals campaign in France.
The FA have since conducted further interviews with Steve Bruce and one other unnamed candidate, but have now, according to the Evening Standard’s James Olley, elected to offer the role to Sunderland’s Allardyce.
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Jurgen Klinsmann had, at one point, been favourite for the role, but the FA’s desire for an Englishman is thought to have ultimately counted against him.
The report adds that Sunderland will now turn to former Everton, Manchester United and Real Sociedad boss David Moyes as Allardyce’s replacement.
The report comes just hours after the FA’s chief executive Martin Glenn had revealed that the governing body’s search for a new manager was nearing a conclusion.
Glenn is heading up a three-man selection panel, alongside FA technical director Dan Ashworth and vice-chairman David Gill, and told the BBC that the panel had spoken to a handful of people.
"We've consulted widely in the game and spoken to a handful of people, and we are getting close." Glenn said.
"The new manager's got to be someone who can inspire people."
Allardyce, 61, began his coaching career in 1991 as player manager of Limerick and has since gone on to enjoy managerial success, most notably at Bolton Wanderers, Blackburn Rovers, West Ham and Sunderland.
Sam Allardyce may not be the fashionable choice as England manager but he is, all things considered, the right choice. International football is a dramatically different beast to club football and, of those considered, Allardyce best suits England’s needs.
Time is a commodity that international coaches are not afforded. Therefore, Allardyce’s pragmatic nature is ideally suited to taking a moderate side – and make no mistake about it, that is exactly what England are – and making the most of them with limited time.
Technical coaches such as Eddie Howe, Brendan Rodgers or even Jurgen Klinsmann may have represented a more progressive appointment, but it takes time to implement their playing systems. Allardyce, on the other hand, has a proven track record of imposing an identity on his teams in an impressively short period of time.
It may not necessarily be pretty, but it is effective. Without being too harsh on Hodgson, England were neither pretty nor effective at the Euros, so Allardyce will more than likely be an upgrade.
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