The reigning La Liga champions are also keen on his team-mate Jan Oblak, and are hopeful of completing a double deal.
Manchester United interest
Gaitain has been linked with a move away from Portugal for some time, but looks likely to make the jump this summer. He helped his side win the Portuguese title this term, although they fell just short in the Europa League, losing in the final against Sevilla.
The 26-year-old played 26 league games for his team last term, and found the back of the net on four occasions. He's long been touted as a potential arrival at Manchester United, while more recently Russian giants Zenit St. Petersburg have shown an interest in him. Tottenham have also been credited with an interest in him.
Marca are reporting that a deal for goalkeeper Oblak is already complete, while negotiations for Gaitan are on-going. The paper say that the fee is likely to be in the region of between €25-28 million - although there is a possibility of Atletico Madrid including a player in the deal in order to lower the fee.
Despite winning the league and making the Champions League final last term, Atletico are in danger of losing their best players, with Chelsea in particular looking to raid Diego Simeone's team.
Manchester United team building
Although it looks likely that they will miss out on Gaitan, Manchester United are certainly in the market for a new midfielder. They've been heavily linked with a move for Arturo Vidal, while a swoop for Bastian Schweinsteiger has also been mooted.
With new boss Louis van Gaal in charge they've already completed two big-money deals for Ander Herrera and Luke Shaw for a combined fee of over £50 million.