Ferguson made his move for van Persie after the Netherlands international, whose current contract expires at the end of the forthcoming campaign, announced earlier this month that he would refuse to sign a new deal with the Gunners.
Premier League champions Manchester City and Italian giants Juventus have also expressed interest in van Persie, but Ferguson tried to steal a march on his rivals with a firm offer.
It is believed Arsenal rejected United's bid because they want at least £20 million ($31.2 million) for the 28-year-old, who scored 37 goals in all competitions last season and won both the Players' and Football Writers' Player of the Year awards.
"Obviously Arsenal have given out the fact that we've made a bid for him," 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."
Van Persie stunned Arsenal coach Arsene Wenger when he declared immediately after Euro 2012 that he wanted to leave the north London team because he felt the club's ambitions did not match his own.
That shock announcement prompted City and Juventus to start the chase for one of Europe's top strikers and United are now firmly in the hunt as well.
Arsenal were expected to announce on Friday whether van Persie would join them on their pre-season tour to Asia but later delayed the decision until Saturday, when they are due to leave the country.
It is widely believed Wenger will opt against including van Persie in a bid to avoid further acrimony.
The van Persie saga threatens to become a carbon copy of those that saw Cesc Fabregas and Samir Nasri sold to Barcelona and Manchester City respectively last year, even though Wenger was desperate to keep both players.