As Sir Alex Ferguson steps up his efforts to find a suitable replacement for Cristiano Ronaldo, Costa's representative, Cesar Bottega confirmed that the two clubs have made contact about the 18-year-old.
But he added that with a reported $21million (£13.5m) get out clause in the player's current contract, not to mention the fee Gremio president Duda Kreoff is holding out for, United look likely to be disappointed.
"There have been some informal conversations [but] I don't believe they can reach those values,' Bottaga said. "Right now, an offer might come in at a maximum of £6.7million. But this would not be enough for [Gremio] president Duda Kroeff.
'At the end of the Brazilian championship, Douglas will be worth at least twice what he's worth now.'
It is not the first time United have shown an interest in signing Costa. Former centre-back Jaap Stam has been monitoring the player for a number of months which prompted speculation of a move back in April of this year.
Meanwhile, United are reported to have agreed to a deal that will see Corinthians' defender Dodo arrive at Old Trafford in January for £5.5 million. With their Brazilian squad members rapidly rising in number, 17-year-old Dodo will be the fifth to arrive from the South American nation in recent years.
Anderson arrived from FC Porto in 2007, twins Rafael and Fabio da Silva both arrived from Fluminese last summer and at the same time Rodrigo Possebon moved from Internacional.
With United providing a stepping stone into the Premier League for such a significant number of Brazilians the club has come under fire from Carlos Alberto - the country's World Cup winning captain from 1970.
According to the Times, Alberto has accused United of "raping Brazilian football" while urging Fifa to step in to protect the country's young talent from the lure of Europe's biggest clubs.
'It is disgraceful United are allowed to rape Brazilian football,' he said. 'The big teams have the money and can take our best players. It is a big problem in Brazil.
'The clubs cannot keep hold of them, if they are good young players, because they can make much more money in other countries. Fifa must do something. They have an obligation to save football. If we could have conditions where we could keep these players in Brazil, it would be much better for us."