The 61-year-old former Bolton, Newcastle and West Ham manager will take over from Roy Hodgson, who stepped down after England's humiliating last-16 exit to Iceland at the European championships in June.
The Press Association said that Allardyce, the long-time frontrunner for the job, will be officially named as the new manager at a Football Association board meeting on Thursday.
Allardyce was in line for the job a decade ago when Sven-Goran Eriksson departed after the 2006 World Cup, but lost out to Steve McClaren.
Hull manager Steve Bruce was also interviewed for the job while Bournemouth boss Eddie Howe and USA coach Jurgen Klinsmann were also reported to be on the FA's shortlist.
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Both Sunderland and Hull publicly called for a new England manager to be hired immediately so they can focus on finding a replacement or get back to planning for the new season, which gets under way on August 13.
England are hoping to have a new boss in place in time for a scheduled friendly on September 1 and the opening 2018 World Cup qualifier in Slovakia on September 4.
However, Sunderland issued a statement late Wednesday to highlight their "anger and frustration" over the ongoing saga.
"Naturally we are aware of the intense media speculation this evening, however, at the present time Sam Allardyce remains Sunderland manager," said a club spokesman.
"We share in the anger and frustration of our supporters and would like to assure them that we are working to conclude the matter in the best interests of Sunderland AFC."
Earlier Wednesday, FA chief executive Martin Glenn had told Sky Sports News: "We're not after a short-term mercenary, someone just to do the job for a couple of years.
Allardyce took over the Sunderland hot-seat from Dick
Advocaat in October 2015 and safely guided the Black Cats to Premier League safety after a season-long battle to avoid relegation.
The northeast club now face the task of finding a ninth permanent boss in less than eight years.
Sunderland are understood to drawn up a list of potential candidates and the names of former Everton manager David Moyes and Burnley's Sean Dyche are at the top of it.
Despite the media frenzy on Wednesday, Glenn had insisted that the FA would not be rushed into a decision.
"We are in a process for the most important job in English football and we are going to get it right," Glenn said.
"Is three weeks a long period of time? If we had done it in three days we would rightly have been accused of knee-jerking.
"We are taking the appropriate amount of time. Of course we are concerned about how individual clubs manage. We aren't blind to that. We just need to make the right decision.