The deal paves the way for Koeman, who has 12 months remaining on this contract, to call time on his two-year stay on the south coast and succeed Roberto Martinez, who was sacked as Everton boss last month following a season of struggle.
Former Holland international Koeman, headhunted by new Everton majority shareholder Farhad Moshiri, is now expected to sign a deal believed to be worth in the region of £6m a year.
The 53-year-old is set to take his brother and Southampton’s assistant manager Erwin Koeman, along with fitness coach Jan Kluitenberg, to Goodison Park with him.
Koeman assumed the Southampton reins from now Tottenham manager Mauricio Pochettino and guided the Saints to seventh and sixth-placed finishes, which represented their best Premier League campaigns.
Southampton will play Europa League football next season after sealing qualification by winning 12 of the final 18 matches last term, which has left former Saints and England forward Matthew Le Tissier querying the foresight of Koeman’s prospective move.
“He may feel he has got a better chance of winning trophies at Everton. I’d be of a slightly different opinion,” said Le Tissier. “I understand they’ve got a new owner and want to splash a bit of cash, but it might not be as easy a job as he thinks.
“Most clubs in the Premier League are pretty wealthy now and can compete in the transfer market. We just have to move on and look to the next man to take us forward again.”
Ex-Barcelona defender Koeman has previously managed at Vitesse Arnhem, Ajax, Benfica. PSV Eindhoven, Valencia, AZ Alkmaar and Feyenoord.