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The Toffees will pay about £5m compensation for the 53-year-old Dutchman, who leaves Saints after two years in charge.
Everton have been without a manager since they sacked Roberto Martinez just before the end of the 2015-16 season.
It is expected Koeman's appointment will be confirmed by Friday.
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Koeman, who has been headhunted by new Everton owner Farhad Moshiri, will sign a deal reported to be worth about £6m a year.
The former Netherlands international, who is on holiday, will take brother Erwin and fitness coach Jan Kluitenberg with him to Goodison Park.
Saints have finished seventh and sixth under Koeman - their best Premier League campaigns.
Martinez promised Champions League qualification when appointed but Everton finished 11th in 2015-16, his third year at the club.
In April, Saints chairman Ralph Krueger said discussions with Koeman over a new deal were progressing "in a very good direction", but added the club was "not in any real hurry".
And last month the former Ajax, PSV Eindhoven and Barcelona defender said he expected to stay with the Saints for the final year of his contract.
Koeman made more than 763 appearances as a player, scoring 253 goals, and won the 1988 European Championship with the Netherlands.
He won the European Cup - now the Champions League - with PSV Eindhoven in 1988 and Barcelona in 1992, scoring the winner for the latter against Sampdoria at Wembley.
He also won four domestic league titles apiece in the Netherlands and Spain.
Koeman has since managed Vitesse Arnhem, Ajax, Benfica, PSV, Valencia, AZ Alkmaar, Feyenoord and Southampton.
He has won the Dutch league with both Ajax and PSV.
Reaction
Former Southampton and England forward Matt le Tissier said he was "disappointed" and "quite surprised" at Koeman's decision to leave, particularly with Saints playing in the Europa League next season.
"He may feel he has got a better chance of winning trophies at Everton. I'd be of a slightly different opinion," Le Tissier told BBC Sport. "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.
"The players' loyalty lies with the football club and hopefully they will kick us on again next season."

The body of India Chipchase, 20, a bar worker, was found in Northampton on Sunday.
Book-keeper Edward Tenniswood, 51, from Stanley Road, faces a charge of murder and has appeared at crown court.
Northamptonshire Police referred itself to the Independent Police Complaints Commission over its "responses before and during" the search.
Read this and other stories from Northamptonshire
A statement issued by the force said: "Northamptonshire Police has made a referral to the Independent Police Complaints Commission to ensure transparency in examining police action and responses before and during the search for India Chipchase.
"We await the decision of the IPCC and will not be commenting further as the criminal investigation is now in the court process."
On Thursday Mr Tenniswood appeared at Northampton Crown Court via video link from Woodhill Prison in Milton Keynes.
When asked if he could hear, Mr Tenniswood told the clerk of the court: "I can hear you, definitely."
After that he spoke only to confirm he could see the Judge Rupert Mayo, prosecutor Mary Loram and his lawyer Derek Johashen.
There was no application for bail and he was remanded in custody.
Ms Chipchase was last seen by friends on Bridge Street at about 01:15 GMT on Saturday.
Her disappearance was reported later that day after she failed to turn up to work at The Collingtree pub and restaurant.
Her body was found at the defendant's terraced house in Stanley Road. A post-mortem examination concluded she died as a result of pressure to the neck.
There will be a plea hearing on the 25 May, and a trial date was set for 18 July, to last for 10 days.
Everton have agreed a deal with Premier League rivals Southampton that paves the way for Ronald Koeman to become their new manager.
[NEXT_CONCEPT]
An investigation into the disappearance of a woman who was allegedly murdered is to be probed by the police watchdog.