The Daily Mail reported Allardyce had signed an 18-month deal and quoted owner Farhad Moshiri as saying: "We need a strong leader, he gives us that."
The former England boss ruled himself out of the running earlier this month, saying he did not feel the Toffees were convinced he was the right man.
But after Everton's pursuit of Watford boss Marco Silva was met with a firm rebuttal from Vicarage Road, the club returned to Allardyce.
This will be the 10th English club managerial position of the 63-year-old's career, with Allardyce most recently having guided Crystal Palace to safety last season.
When he left Selhurst Park in May, Allardyce said he had no ambitions to take another managerial job but the chance to turn around the fortunes of the Merseyside club has apparently lured him back.
Everton are 17th in the table, two points ahead of West Ham in 18th.
David Unsworth has been in temporary charge since Koeman was sacked on October 23 and the under-23s boss had made clear his desire to take on the job permanently.
But during his seven-game tenure Everton have lost five and won just one.
Everton invested heavily following last season's impressive seventh-place finish.
Allardyce has previously led Palace and Sunderland to safety from precarious situations.
His achievements at the Stadium of Light earned him his dream job with England but that turned sour after only one match, with Allardyce stepping down in September 2016 following a newspaper sting....