Bradley is on the look out for a new central defender in the new year after admitting that Ashley Williams has not been properly replaced since he joined Everton and Birnbaum's name has been thrown into the frame.
It comes as no surprise to see American players linked with a move to the Liberty Stadium, but Bradley has shot down the rumours suggesting Birnbaum is a target.
Swansea fans would not be blown away by the link and Bradley himself has suggested Birnbaum may not be ready for the Premier League.
"I know Steve pretty well, although I've never coached him," said Bradley. "He came into the league at DC United, he has done pretty well there and has then been called in at various times to the US national team.
"But there's been no contact from me at this point with Steve. I know that there’s been speculation at times in the last year or so that he was considering opportunities outside MLS.
"I don't know where he stands. It's like any situation around the world. You have players in MLS who at a certain point are getting to the end of their contract and then there's rumours about that. I know him but he's not been a player we've had contact with or put on any kind of shortlist.
"It's a big step. Is he ready for the Premier League? I'm not sure."