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.
This will now be of little concern to the United boss, who has consistently stated how much backing he has received from the Glazer family.
Despite the presence of England Euro 2012 star Danny Welbeck and Javier Hernandez in his squad, in addition to Berbatov, who has been widely tipped to leave Old Trafford, Ferguson made van Persie one of his main summer targets.
The Red Devils initially lodged a bid for the 29-year-old last month, but this was turned down by the Gunners amid further interest from Manchester City and Juventus.
However, City never followed up an initial inquiry, while van Persie made it clear he was not interested in a move to Serie A.
Having taken the unusual step of confirming a bid for van Persie on United's pre-season tour of South Africa, Ferguson is then said to have intervened in a deal that was in danger of stalling by speaking directly with Arsenal manager Arsene Wenger.
Although Wenger did not want to sell his star man to a rival club, he realised the Gunners were in a tricky situation, with supporters turning against their former favourite during a pre-season encounter with Cologne on Sunday.
United fans will certainly be thrilled by the potential of Wayne Rooney partnering last season's Golden Boot winner.
Van Persie scored 37 goals in all competitions, enough to land him both the PFA and FWA player of the year awards.
In addition, the Holland international managed 15 assists.
If there is a risk to the transfer it is van Persie's patchy injury record.
His 48 games last season was by far the most during his time at Arsenal.
In only three of his other seven seasons has he featured on more than 35 occasions.
Ferguson will presumably believe he has the resources that will ensure not too much is demanded of van Persie, especially as Welbeck is believed to be on the brink of agreeing a new contract.
It also brings to an end one of the summer's most long-running transfer sagas.
Van Persie is expected to fly straight to Manchester to complete his medical tomorrow after he was left out of Holland's friendly with Belgium in Brussels.