Wolves had previously failed with a bid of £12m for the winger, but Nuno Espirito Santo is determined to get his man. Traore is understood to have a release clause and Wolves may have to meet it in order to prise the 22-year-old away from Tony Pulis.
The Telegraph claim a fee of £18m would be enough to secure the former Barcelona winger's services, breaking Wolves' previous record transfer of £15m for Ruben Neves.
The move is supposedly progressing after the former Spain Under-21 international partially dislocated his shoulder in a pre-season friendly. However, the former Aston Villa man has now resumed training, making his transfer to Wolves a real possibility.
Although a move to the Premier League may seem promising, Boro boss Pulis has warned Adama against moving to Molineux this window. Pulis is quoted in the Express and Star as telling a fans' forum: "I've told him, I think another season with us and he'll get a top six club coming in, not just Premier League clubs.
"If he plays as well as he did last year and even improves, he could have top six clubs coming in for him." The Welshman added: "He has to make the decision, listen to those around him, the people he respects.
"Then I will give him and the club will give him the respect, and everybody has to give him the respect he deserves as a human being to do what he feels is right for Adama Traore. But we're doing our best to try and keep him here."
Wolves will host Everton in their first season back in the Premier League. The Championship winners are expected to do well this season after spending big in the transfer window so far.