Dubbed the "new Ronaldinho", United have been courting the 18-year-old for a number of months now and had hoped his club, Gremio, would permit a trial period with a view to a permanent move.
Costa - who has attracted the interest of Liverpool, Inter Milan, Real Madrid and Villarreal - is valued at £18m by Gremio, a fee United are unwilling to pay, but it is thought they would offer an initial £6.7m, which would then rise depending on appearances.
Former United centre-back Jaap Stam has been monitoring Costa on the club's behalf but Sir Alex Ferguson is keen to see the young player perform first-hand before committing to a contract.
Gremio, though, have dismissed such an idea, and club president Duda Kroeff said: "He's not at the stage where he'd go through such a situation."
Meanwhile, United's hopes of signing Karim Benzema have been handed a boost after Lyon made a £15m move for Porto forward Lisandro Lopez.
Lyon president Jean-Michel Aulas has gone on record as saying that the club do not want to sell Benzema.
However, amid keen interest from United, Real Madrid and Barcelona, the French outfit have opened the door for Benzema's departure by bidding for Lopez.