The three clubs have all declared their interest in buying want-away striker Van Persie but have so far failed to meet the Gunners' valuation of the 28-year-old.
"Obviously Arsenal have given out the fact that we've made a bid for him," Sir Alex Ferguson told MUTV.
"We try to keep business as close as possible to us until the moment we conclude a deal. There's no point talking about something that might not happen.
"There are other clubs interested in the player. We've shown an interest and that's where we're at at the moment."
United, City and Juve are said to be happy to pay somewhere in the region of £15 million but Arsenal are thought to be demanding in excess of £20m for their star player.
The Gunners are said to be prepared to hold out for more than £25m for Van Persie, who they appear to have all but given up trying to convince to sign a new deal at the Emirates Stadium to replace the one that expires next summer.
According to widespread reports, manager Arsene Wenger has opted against including Van Persie, who this month went public with his desire to quit the club, in the club's pre-season tour.