The 26-year-old Argentine international - who is rated at around £15million - is also attracting interest from Serie A champions Juventus, as well as a number of other top European clubs.
Red Devils boss Sir Alex Ferguson has identified Garay as a long-term replacement for Rio Ferdinand and Nemanja Vidic, with both players struggling for form and fitness at Old Trafford.
Ferdinand, 34, will be out of contract at the end of the season, and there is still some doubt over whether or not he will sign a new deal, while club captain, Vidic, has been in and out of the side due to persistent injury.
Manchester United scouts were present at Benfica games against Sporting Lisbon and Olhanense earlier this season, running the rule over the South American stopper, and they are also believed to have taken in the Portuguese Liga leader's top-of-the-table clash against FC Porto at the weekend.
A 2-2 draw maintained the Lisbon club's three point lead at the summit, but with Benfica under financial pressure at the moment, their title challenge might be hampered by the possible sale of star players like Garay, who they need to get off the wage bill.
Real Madrid are also entitled to half of the fee that the centre-back is eventually sold for, following the Argentinean's exit in 2011, after he failed to make the grade at the Santiago Bernabeu.
That will prompt Benfica to hold out for as much money as possible, and could even see them play Manchester United off against other interested parties, before selling to the highest bidder.