The Swans have conceded 26 goals in just 13 Premier League games this season and Bradley is eager to add to their defensive options in January.
DC United's Birnbaum, 25, is understood to be the Swans' top target, with Besler of Sporting Kansas City an alternative option should a move for Birnbaum fail.
A deal to bring in Birnbaum should be fairly straightforward, however, as DC United are owned by Swans co-owner Steve Levein.
Swansea will reportedly try and snare the defender on a temporary deal in January as Bradley bids to strengthen his back line.
Birnbaum has nine caps for the USA to his name while 29-year-old Besler has appeared 39 times for the USA and has spent his entire career with Sporting Kansas.