Keane progressed through the Manchester United academy before joining Burnley in a reported £2.5million deal in January 2015 but any potential suitor will have to pay a significantly higher fee to land the 23-year-old fee in the upcoming window.
Liverpool boss Jurgen Klopp is understood to have made improving his defensive options a priority despite his side's fine start to the season.
The Reds are currently fourth with 16 points but they have conceded more Premier League goals than any other top 10 side and are likely to bring in at least one new defender in January.