The good news looks set to continue at Goodison Park.
Everton are keen to tie down the England internationals, a report claims
Talks to improve Barkley's reported £60,000 have long been planned, according to Bascombe, who adds that Koeman's imminent arrival will increase the Toffees' chances of keeping hold of Stones.
Unlike his team-mate, Stones has previously expressed a desire to leave Goodison Park, with Chelsea seeing numerous bids turned down last summer.
Stones struggled under previous boss Martinez
And Bascombe claims that had Roberto Martinez remained in charge, the former Barnsley centre-back would have reiterated his stance, amid interest from the Blues and both Manchester clubs, United and City.
However, thanks to the influence of Koeman, one of the game's finest ball-playing defenders, Everton are said to have grown increasingly confident of keeping the 22-year-old.
Ambitious Moshiri is making a big impact on Merseyside
In luring Koeman from sixth-placed Southampton, new investor Farhad Moshiri has shown a level of ambition that, Bascombe writes, will be "applied to retaining" key personnel and building upon them.
Reports claim the Dutchman is set to receive a £100 million transfer warchest, having doubled his Southampton wages to £7 million-a-year.
Koeman is said to have doubled his wages