The 21-year-old attacking midfielder has started all 13 of his side's matches in the Premier League this season, scoring five times.
Kilbane, who played for Everton between 2003 and 2006, has been hugely impressed with the England man's performances and says he is already talented enough to play at the highest level.
He told The Daily Star: "Is Barkley good enough for United or Chelsea? Yes he is.
"If you are talking about playing for one of the top sides, of course he has that ability.
"He has got power, he has got pace, he is two footed. He has that rare ability to change a game and bring it to life.
"When you watch him play, he can glide past players, he can pick passes out."
Kilbane feels Barkey has shown a huge improvement in 2015-16 and is becoming more consistent with experience.
"He has taken quite a bit of stick from the Everton fans over the last couple of years," added the former Republic of Ireland international.
"A young lad like that is going to have inconsistent spells and times where he is not going to be in the match. But this season we are starting to see his skills and his level of performance has been outstanding.
"He has been involved with so many good things. If it is not a direct assist he has played a big part in the lead up to Everton's goals quite a lot this season."
