The report states that the 24-year old defender has agreed personal terms in principle with the Anfield club, but he is waiting on whether Jurgen Klopp's side qualify for next year's Champions League to make his decision.
The United academy graduate was sold from Old Trafford to the Clarets in 2014 for £2.5million after then manager Louis Van Gaal deemed him surplus to requirements.
Since then, the Stockport-born lad has blossomed into one of the country's top class defenders. Keane, who made his full England debut against Germany in March, has had a stellar Premier League season, which has been recognised with him being added to the short-list for the PFA Young Player of the Year award.
Both clubs have expressed their desire to add to their centre half ranks and Keane is believed to be their number one priority.