The England international completed a cut-price £15m move to the defending Premier League champions from his boyhood club Everton last week.
Barkley has put pen to paper on a lucrative four-and-a-half-year contract at the west London side in a bid to kick-start his ailing career.
The 24-year-old attacking midfielder struggled to maintain his progress at Everton following some impressive performances under Roberto Martinez.
Indeed, Barkley appeared to lose his way under the Spanish coach’s successor Ronald Koeman, prompting speculation about his long-term future at the Merseyside club.
The England star appeared to struggle under the pressure of being Everton’s hometown hero, with his lack of discipline being cited as a big flaw in his game.
Former Chelsea defender Burley warned Barkley that there will be no hiding place at Stamford Bridge following his move to the Premier League champions.
“His problem now is he is in the limelight,” Burley told ESPN2.
“The real limelight. Getting a game in this Chelsea side is going to be difficult on a regular basis.”
The former Everton midfielder has netted 21 times in 150 games in the Premier League.
Barkley will have to compete alongside Cesc Fabregas, Tiemoue Bakayoko, Danny Drinkwater and N’Golo Kante.
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