Barkley was named in England’s squad for the international tournament, but he was left on the substitutes’ bench for every match the Three Lions played.
England lacked a creative edge in every one of the matches they played in France, and Redknapp feels that Roy Hodgson’s omission of Barkley was an error.
"I would have persevered with Ross Barkley and even thrown him on against Iceland the other night," Redknapp said.
"At half-time, I’d say to him: “Okay, I didn’t start you but now’s your chance. Come out here, change this game for us. We need to show what you can do — you are different class. Run the game, cement your place in the team and take us into the quarter-final.” Hopefully Barkley would feel 10ft tall."
Barkley enjoyed his most productive season in an Everton shirt going into the European Championships, after claiming 12 assists.
The 22-year-old was forced to watch for the sidelines, however, with the likes of Dele Alli, Jack Wilshere and Adam Lallana all starting ahead of him.
England’s Euro 2016 campaign ended in disastrous fashion, with the Three Lions crashing to defeat against Iceland.
Hodgson has now lost his job, with Redknapp one of the candidates tipped to takeover the national side.
As well as Barkley, Everton’s John Stones was in England’s squad, but he also failed to make it on to the pitch.