Keown, who had a spell at Everton in the late 1980s and early 1990s, has claimed that Barkley took a backward step during the 2015-16 campaign.
The former England international rates the 22-year-old attacking midfielder highly, though, and believes that he needs to step up his game during the 2016-17 season.
Everton's Ross Barkley in action
“Ross Barkley won’t have enjoyed being pushed aside by Dele Alli at international level — he remains a tremendous talent but looked as though he took a backward step last season,” Keown wrote in The Daily Mail regarding the Everton player who has got a point to prove.
“He’s probably Everton’s most gifted player — his technique and dribbling ability are exceptional — but now is the time for him to mature and improve his decision making and composure.”
Barkley is one of the best and most promising young players in England, and the attacking midfielder will be looking to take his game to the next level this season.
Everton's Ross Barkley
The England international attacking midfielder is hugely talented, but he needs to find consistency.
The 22-year-old scored eight goals and created 55 chances in 38 Premier League matches for Everton last season.
Keown has also given his prediction on how the Toffees will fare in the Premier League this season.
Everton have a new manager in charge, Ronald Koeman, and the fans will expect the team to challenge for a top-six finish in the league table.
Former Aston Villa man Keown expects the Merseyside outfit to improve dramatically this season.
“I think Everton will dramatically improve this season — they have some excellent talent and they underachieved last term,” Keown wrote in The Daily Mail. “There are lots of good teams ahead of them, but I could see them qualifying for Europe.”
Everton endured a disappointing 2015-16 campaign and failed to finish in the top 10 of the Premier League table last season.