The Dutch striker, voted the league's player of the year last season, will travel to Manchester on Thursday to discuss personal terms and complete a medical ahead of the proposed move to Old Trafford.
No financial details were immediately available from the clubs. Media reports said United would pay £24 million for Van Persie, who is poised to sign a four-year contract with them.
The 29-year-old, who had also attracted interest from the Premier League champions Manchester City and Juventus, said last month he would not extend his contract at Arsenal. He had one year left on his deal after eight years with the club.
Van Persie scored 37 goals last season as Arsenal finished third in the league and reached the last 16 of the Champions League. But the club have not won a trophy since the 2005 FA Cup final defeat of United.
At Old Trafford Van Persie would link up with England striker Wayne Rooney as Alex Ferguson's team try to take the title back from their city rivals. United have made only one major signing since the end of last season, Japan midfielder Shinji Kagawa joining from German champions Borussia Dortmund.
Arsenal manager Arsene Wenger was keen to avoid a repeat of the tumultuous start to last season when protracted transfer speculation swirled around their then captain Cesc Fabregas and his fellow midfielder Samir Nasri.
Fabregas moved to Barcelona in mid-August and Nasri was sold to Manchester City nine days later. Before the end of the month Arsenal had suffered an 8-2 hammering by United at Old Trafford.