The Daily Mail understands that Rodgers will turn to Williams as a replacement for Daniel Agger, who looks likely to sign for Manchester City.
Agger was absent from Liverpool's Europa League tie against FC Gomel on Thursday to add strength to the rumours linking the Dane with a move to the reigning Premier League champions.
Rodgers had promised not to upset his old club by trying to sign Swansea players but has already been consistently linked with a deal for Joe Allen this summer.
The Reds boss is a long-term admirer of Williams and would like to bring the Welsh international to Anfield.
Liverpool begin their Premier League campaign against West Bromwich Albion on Saturday, August 18.