Keane, 22, came up through the ranks at Manchester United, but having failed to break into the Red Devils' first team, the defender left Old Trafford last summer, joining Burnley initially on loan before a permanent deal in January.
The centre back has since impressed on a regular basis for Sean Dyche's Clarets, and Premier League clubs are swarming to potentially offer him a return to the top flight.
The Sun report that Everton are interested in signing Keane when the transfer window re-opens in January, whilst Newcastle United boss Steve McClaren - who signed Keane on loan whilst managing Derby County - is reportedly looking to reunite with the defender on Tyneside.
Everton may have signed Argentine Ramiro Funes Mori from River Plate this summer, but manager Roberto Martinez is still a little short in central defence, having released Sylvain Distin and Antolin Alcaraz at the end of last season.
Meanwhile, Newcastle snapped up Chancel Mbemba this summer, though the advancing age of Fabricio Coloccini means that a younger option may be on McClaren's shopping list.
Keane's displays in the Championship are unsurprisingly catching the interest of Premier League clubs, but a reunion with McClaren may be appealing having enjoyed that loan spell under him at Derby County back in 2013.