After leading Saints to their best Premier League finish in back-to-back seasons, the 53-year-old had started negotiations about extending his stay at St Mary's.
Former Barcelona defender Koeman had a season remaining on his three-year deal, but the Dutchman now looks set for a move to Goodison Park.
Press Association Sport understands a compensation fee of around £5million has been discussed and a resolution could be reached as soon as Monday.
Rob Jansen, who recently replaced Guido Albers as Koeman's agent, told Voetbal International on Saturday that they had "reached an agreement with Everton" and "now it is up to the clubs".
Koeman is understood to have been the first choice of the club's new major shareholder Farhad Moshiri, although Sevilla manager Unai Emery is believed to have been preferred by some others at Goodison Park.
The Spaniard, fresh from winning a third straight Europa League, is among the bookmakers' favourites to succeed the Dutchman at St Mary's, along with the likes of Eddie Howe, Frank de Boer, Andre Villas-Boas and Manuel Pellegrini.
Source : PA
Source: PA