The Daily Mail is reporting that Everton, Bournemouth, Aston Villa, Stoke City, Norwich City and Swansea City are all keen on the former Manchester United youngster - who joined The Clarets last September.
Everton are in need of another centre-back option. Tyias Browning is set for a lengthy spell on the sidelines with a knee problem and Phil Jagielka is also currently crocked - leaving centre-back options bare.
Bournemouth have struggled at the back in the Premier League this season and would benefit from a reinforcement.
Aston Villa's defenders have failed to impress this term, with Micah Richards needing a new partner beside him.
Stoke, Norwich and Swansea could all also do with a recruit at the heart of their defence for the rest of the season.
Keane has been in great form for Burnley this season - playing every minute of their Championship campaign and even pitching in with three goals.
There is no doubt he would be a welcome addition to any one from Everton, Bournemouth, Aston Villa, Stoke City, Norwich City or Swansea City - but with Burnley in a promotion hunt themselves he might just decide to stick around.