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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Chelsea have struggled to build momentum in the opening stages of the campaign and are currently looking over their shoulders at the relegation zone.
Who of Europe’s top strikers will move to England this summer?