The Red Devils are said to be long-term admirers of the Argentine international and were said to be keen on bringing the 25-year-old to the Premier League over the summer.
Gaitan decided to stay at Benfica for the opening six months of the season but has since admitted he could follow Nemanja Matic to England.
However Manchester United will have to stump up the same amount of money as Chelsea paid for former Benfica midfielder Nemanja Matic, which is believed to be around the £21million mark.
Reports claim French moneybags Paris Saint-Germain will rival Manchester United for Gaitan's signature as Laurent Blanc is keen on making a move for the highly-rated attacking midfielder.
Manchester United boss David Moyes - who has sent scouts over to Portugal to watch Gaitan - is also said to be keeping close tabs on Benfica centre-back Ezequiel Garay.