The Benfica playmaker has been a long-term target for Sir Alex Ferguson, and has also attracted interest from Tottenham and Anzhi Makhachkala in recent months.
But Inter Milan are ready to sanction a £12m offer for Gaitan, with Inter midfielder Ricky Alvarez also moving in the other direction.
Gaitan was signed by the Liga Sagres leaders from Boca Juniors three seasons ago as a replacement for the Real Madrid bound Angel Di Maria.
And the report claims many onlookers believe the 25-year-old Gaitan has the potential to be even better than Di Maria.
He has six caps for Argentina, and three goals and three assists from 20 appearances for table-topping Benfica this term.
But United and Tottenham will have to be quick if they still want to land the midfielder, as the report claims Inter Milan have talks planned for next week in Lisbon.
Inter have struggled in Serie 'A' this term, and are looking to overhaul the squad this summer to mount a renewed title challenge next season.
Tottenham are also hoping to qualify for the Champions League, and have identified Gaitan as a player with the necessary quality and experience of European competition.
But Spurs signed Gylfi Sigurdsson and Clint Dempsey last summer, and both operate in a similar space to Gaitan.
United meanwhile have question marks over the future of Nani and Javier Hernandez, and could be in the market for a forward-thinking player at the end of the season.