The Welsh club are after the former Crystal Palace and Spurs man.
Routledge had seemed set to join QPR, after impressing whilst on loan at Loftus Road last season, but the R's were unable to strike a deal and that put the 26-year-old's future on hold.
Cardiff City and Nottingham Forest are understood to have both made approaches for the winger, but the lure of the top flight may put Swansea in the box seat.