With the 23-year-old intent on leaving Arsenal next summer, he knows there will be no shortage of admirers.
Bruce told the Newcastle Journal: "I have made no secret we would love to sign him.
"However, it is going to be one of those again, the better he does, certainly internationally, (the more) people are looking at him.
"I spoke to (Denmark coach) Morten Olsen on Thursday and he was delighted with his performances and, I have to say, so am I for us.
"He has done terrifically well, so let’s hope he keeps doing that."