The 28-year-old midfielder, who cost 28 million pounds when he joined United from Italian club Lazio in 2001, is now set to hold contract talks with the big-spending London club.
In a statement to the London Stock Exchange on Wednesday, United confirmed they had accepted Chelsea's bid "subject to contract and player medical".
Chelsea chief executive Trevor Birch told the club's website: "They have accepted our bid of 15 million pounds.
"And now providing he wishes to speak to us we can proceed with negotiating personal terms and putting him through a medical."
Chelsea will be anxious to agree personal terms with the Argentine quickly as the deadline for registering players for the Champions League third qualifying round is 1000 GMT on Thursday.
United said that 2.5 million pounds of the 15 million fee was "conditional on the performance of the Chelsea team over the next four seasons".
Veron would be the sixth high-profile signing made by Chelsea since the takeover by billionaire Russian businessman Roman Abramovich last month.
On Tuesday, Chelsea agreed a fee of 6.6 million pounds with West Ham United for promising young England midfielder Joe Cole.
That came after they had signed up Ireland winger Damien Duff from Blackburn Rovers for 17 million pounds, West Ham defender Glen Johnson (up to 6.0 million), Cameroon midfielder Geremi (6.9 million) and England defender Wayne Bridge (7.0 million).
Veron's departure from United would end a disappointing two years for the Argentine, who struggled to produce his best form during most of his time at Old Trafford.
The former Boca Juniors, Sampdoria and Parma player helped United to the league title last season but injury and form restricted his appearances in a fiercely competitive United midfield.
United manager Alex Ferguson was forced on several occasions to defend Veron against media suggestions that the Argentine did not fit into his preferred system of playing.
