The Sun reports that United boss Sir Alex Ferguson is hoping to complete a £22million deal before the Premier League starts this weekend.
Van Persie is desperate for a move to Old Trafford to be completed.
Arsenal were demanding £25million for the striker, but are now prepared to settle for less.
Van Persie scored 37 goals in all competitions for Arsenal last season.
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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.