Barkley will have just one year remaining on his contract at Goodison Park this summer and, according to The Daily Telegraph, the England international is high on Mauricio Pochettino's shortlist as part of a bid to bolster his Spurs squad.
The 23-year-old initially struggled under Koeman but his form has drastically improved over the last couple of months having benefited from the tough love shown by the Dutchman.
While Tottenham may harbour hopes of luring the Wavertree-born star down south, Koeman's recent comments suggest a move away that may have looked on the cards earlier on this season is now completely off the table.
"Of course we'd like to keep Ross. I think he is the type of player you need in your team to create chances and score goals," Koeman told Sky Sports.
"Sometimes you have fights with players because I like to see more productivity and it's difficult for players to accept criticism, but I think that's necessary to get the best out of the player.
"I have a lot of communication with Ross. He really is a good guy. He likes to improve and he'd like to stay at Everton.
"I think he's changed his football in a positive way and that's really important for himself and the team."
While Koeman sees Barkley's future on Merseyside, Tottenham could still test their resolve with a bid in the summer. The Telegraph suggest the attacking midfielder is one of three players the club are possibly looking to sign, with Pochettino having already held talks with club chairman Daniel Levy.
The Spurs coach feels the club must add to their squad if they are to keep up with their rivals' rampant spending. The report suggests Crystal Palace winger Wilfried Zaha, who Spurs were intensely linked with last summer, and Burnley striker Andre Gray are on Pochettino's radar.