The Reds, 15 points clear at the top of the Barclays Premier League, meet the London side at Stamford Bridge on Monday in an FA Cup sixth-round replay. The winner will secure a Wembley semi-final against Manchester City on Sunday 14 April.
Not since 1999, when Sir Alex Ferguson's side won the Treble, have United lifted both the league title and the FA Cup in the same season. But van Persie, who arrived at Old Trafford last summer determined to win silverware with his new club, admits he's harbouring thoughts of a historic Double.
"We want to win the Double and we took a big step in the league [on Saturday against Sunderland]," he told MUTV. "But we want to go for both. Now we have to look ahead to the FA Cup tie against Chelsea. It's going to be a big test but I'm quite confident we can play well at Stamford Bridge and get a good result."
Since the Reds' last FA Cup success in 2004, United have lost two FA Cup finals, against Arsenal in 2005 and Chelsea in 2007. Van Persie actually played in that 2005 final, a match the Gunners won on penalties after United dominated for 120 minutes but failed to score.
"It's been nine years since United have won the FA Cup," the Dutchman said. "It's important to us. We have a big chance of reaching the semi-final against City. I don't want to look too far ahead but if we beat [Chelsea and then] City we have a massive chance of winning the competition."
Monday's game against Chelsea comes less than 48 hours after both sides' Saturday league fixtures. As a result, van Persie warns fans shouldn't expect a classic encounter.