The announcement comes after an intense period of negotiations and is subject to Dutch striker Van Persie passing a medical and agreeing personal terms.
"Manchester United is pleased to announce it has reached agreement with Arsenal Football Club for the transfer of Robin Van Persie," said a club statement. "The deal is subject to a medical and the agreement of personal terms. A further announcement will be made in due course."
It is not anticipated there will be any problems with either remaining part of the deal.
And providing it goes through without a hitch, Van Persie could face Everton in United's opening Premier League game at Goodison Park on Monday.
No fee has been confirmed by United, although it is believed to be around the £20million mark, with further payments to Arsenal depending upon success in the future.
It has been suggested Van Persie will sign a four-year deal.
The transfer represents the biggest deal United have done since Dimitar Berbatov's club record £30.75m move from Tottenham in 2008.
It comes at a time when further questions are being raised about the finances available to Sir Alex Ferguson.
Source: PA
Source: PA