Reports on Tuesday night had suggested that the deal was close to being completed, and the Premiership champions confirmed the news to the Stock Exchange on Wednesday morning.
The former Lazio man will still need to agree on personal terms with Claudio Ranieri's side, but that should not be a problem for the deep-pocketed Roman Abramovich.
£2.5 million of the fee is also dependent on Chelsea's performances, and Veron is due at Stamford Bridge to complete a medical on Wednesday.
Blues chief executive Trevor Birch confirmed the news to the club's official website and said that personal terms would be discussed with Veron as soon as possible.
"They have accepted our bid of £15m," Birch told the club website.
"And now providing he wishes to speak to us we can proceed with negotiating personal terms and putting him through a medical."
Veron's agent Fernando Hidalgo admitted that the Argentine ace was happy to move to Chelsea and that he was set to sign a four-year deal should he agree terms and pass his medical.
"The player will arrive in London at midday. He's going to take his medical tests and he will sign a four-year contract," said Hidalgo.
"He is happy. It has been a club [United] decision, and he has accepted it. He is ready to do the best he can with this new opportunity at Chelsea."
Hidalgo also revealed that although Veron never wanted to leave Old Trafford he was proud to be joining Chelsea.
"When decisions are made, everyone has to follow the consequences,"added Hidalgo.
"Seba is proud to join Chelsea - but it was not his intention to leave the club [United]."
The news comes only hours after confirmation that Chelsea had also agreed a fee with West Ham for Joe Cole, and should both deals go through it would take the club's summer spending spree up to £58 million.
The deal represents a big loss for United after paying Lazio £28.1 million for Veron two years ago, and Old Trafford fans will want to see the cash used for new players after seeing both Veron and David Beckham leave the club.
