Koeman is expected to be confirmed as Everton manager in the coming days having been tempted to quit Southampton by majority shareholder Farhad Moshiri.
Moshiri is ready to give Koeman significant funds to improve the squad at Goodison Park but is also ready to splash the cash on keeping the existing team intact.
Barkley's existing £60,000-a-week deal has two more years to run but the England midfielder looks set to be rewarded with a new long-term contract with significant payrise.
The 22-year-old has never shown any inclination to leave his boyhood club so agreeing terms shouldn't be too complicated.
Keeping John Stones at the club may be tougher, though Moshiri's arrival and Koeman's imminent appointment offers Everton hope.
Had Roberto Martinez still been in charge Stones would certainly have left, with Chelsea and both Manchester clubs rumoured to be interested.
However, one of Koeman's first jobs will be to talk to Stones and try and convince him to stay at Goodison Park. The Dutchman would be the perfect mentor for the former Barnsley man having been a fine ball-playing centre back himself.
Stones would also be in line for a significant payrise should he commit to the club and combined with Moshiri's ambitious vision for the future it may well be enough to persuade him to stay.