The Belgian worldwide was initially expected to re-join former club Chelsea but United moved in to pole position after reports emerged late last week [via the BBC] that the Red Devils had agreed a fee for Lukaku's signature.
But elsewhere there are reports the Spanish club are yet to make an opening bid for the forward.
And despite winning the title with the Blues last term, Costa angered Antonio Conte by trying to force a move away from Stamford Bridge in January.
With Costa duly making that message public, this has effectively weakened Chelsea's negotiating position in the process.
The same article states that Costa has told some of his Chelsea team-mates that he is confident a return to Atletico will materialise in the summer transfer window.
Chelsea remain adamant they will not be forced into accepting a deal with which they are displeased however with Costa only demanding a reunion with Atletico, they might be unable to command a high price for their leading scorer last season.
His revelation naturally alerted a host of teams, notably his former club Atletico Madrid, whom many expected him to join even if that means not playing for the first half of the 2017/18 season due to the Spanish club serving a transfer ban.
It has been reported that Atletico may sign Costa and then loan him to a Chinese club.
Atletico Madrid have been closely linked with the Spanish worldwide but Besiktas have emerged as a surprise destination.
Costa scored 27 goals in all competitions last season under Conte, cementing himself as one of the best strikers in the Premier League for another season.