Brady, 20, has impressed Hull fans and manager Steve Bruce during his second spell on loan to the Championship side, having helped Hull move into the top four in the league table on the back of a run of three straight wins.
Brady's loan deal is due to expire next month and Hull, pushing hard for a return to the Premier League, could offer Bruce the funds to sign Brady on a long-term deal.
Bruce has already stated that he wants to extend the loan deals for stars like Brady and Sunderland man David Meyler beyond January, but City could test United's resolve with a formal offer.
It's thought that any deal for Brady could cost the club £2million up front, but United boss Alex Ferguson may decide to keep the Dubliner on board at Old Trafford and give Brady the chance to force his way into the United first team.
The 20-year-old made one appearance for United this season, in a League Cup win over Newcastle United.