Barkley's future was cast into doubt this past summer after he failed to agree to a new deal at Everton amid the arrivals of Wayne Rooney, Davy Klaassen and Gylfi Sigurdsson.
Speaking about his move, Barkley told the club's official website: "I'm overwhelmed, I'm looking forward to it and I'm really excited to get started".
Speculation had been rife that Barkley even travelled to Chelsea for a medical only to change his mind at the last moment, news he denied.
Toffees chief Allardyce has wished the 24-year-old well and is relieved the speculation can now stop.
Barkley is still recovering from his long-term injury which has ruled him out for the whole season so far, and Osman says the midfielder must prove his worth when he eventually returns to action.
A move to Chelsea on transfer deadline day in August did not materialise but Barkley has now moved to Stamford Bridge for a cut-price fee.
Kevin Durant and Steph Curry Lead Western Conference All-Star Voting
The All-Star Game captains and starters will be announced on TNT Jan. 18, and the reserves will be announced Jan. 23. There are still 11 more days left in the All-Star voting, which will conclude on January 15 at 11:59 p.m.
Conte is desperate to reinforce a squad that has left him 16 points behind Manchester City in the Premier League.
'We are talking about an English player, a young player, with good prospects for Chelsea, ' Conte said. He's very strong physically, good technique.
Conte urged caution over expectations for Barkley.
'He's very young. He has a lot of space for improvement.
"I think sometimes when you are at your home club, you can be there too long and you have to learn the different levels of the game. I have to still see him and then we have the opportunity to work together and understand how much time he needs [until he makes his debut]".
"He had a bad injury, he had surgery and he hasn't played for seven months".