Williams has been a star of the Swansea side which has reached the final of the Capital One Cup.
The Daily Mail report that Brendan Rodgers is keen to bring his former player to Anfield, though Arsenal may prove an obstacle to those hopes.
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.