The transfer will be completed as soon as the Dutch striker passes a medical and agrees terms with United.
An official club statement said: "Manchester United are pleased to announce an agreement with Arsenal Football Club for Robin van Persie.
"The deal is subject to a medical and the agreement of personal terms. A further announcement will be made in due course."
Van Persie, 29, netted 37 goals in all competitions last term, 30 of which were in the Premier League.
His capture will be music to the ears of United boss Alex Ferguson, who has long been an admirer of the Holland international
...RVP is our Wayne man
ALEX FERGUSON landed Robin van Persie last night for £22million. And he promised him the chance to form a dream partnership at Manchester United with Wayne Rooney.Van Persie will travel to Old Trafford today for a medical after agreeing terms of just over £200,000 a week.
As SunSport said on Tuesday, boss Fergie hoped to complete the transfer before the big kick-off this weekend.
Van Persie, 29, who refused to sign a new deal at Arsenal, has been told he will be played at United as an out-and-out striker. Rooney will operate in a free role behind him.
Fergie had feared Manchester City could still land RVP. City boss Roberto Mancini had insisted that the club were no longer in a position to secure him but United believed this was not the case.
Rooney was also delighted at the news and took to twitter to taunt celebrity Arsenal fan Piers Morgan.
He said: “Arrrr, poor Piers. what’s up now, big man?” Morgan responded by calling Roo “the No 2 striker at Old Trafford”.
United will pay £15m cash up front with the fee rising to a guaranteed £22m. But, depending on other add-ons, Arsenal could still land £24m. Gunners boss Arsene Wenger said: “It’s sad to lose a player of his quality. He had a year left on his deal so we didn’t have the choice.”
When asked if he would spend on new players, Wenger said: “We’ve already recruited to replace him.”
In a club statement, the Gunners said: “Van Persie will travel up to Manchester on Thursday so he can agree terms and complete a medical ahead of the proposed transfer.”
A spokesman for United added: “We are pleased to announce we’ve reached agreement with Arsenal for the transfer of Robin Van Persie.” United midfielder Tom Cleverley was told the news after playing in England’s 2-1 win against Italy last night.
And he said: “He’s a great player — he can only make us strong.”