Van Persie joined the Gunners from Feyenoord in 2004 but was also tracked by United while he was at the Dutch club.
"I remember going to see Van Persie play for Feyenoord reserves, way, way back," said Ferguson.
"We were definitely interested, but the day we went to see him he didn't play the full game.
"He was either sent off or he walked off, I can't remember which. We ended up leaving it, we didn't make a move for him."
Ferguson will come up against Van Persie when United meet Arsenal at the Emirates Stadium this afternoon.