The newspaper claims Hart has been identified as a potential replacement for Stoke's Jack Butland, for whom the Potters are braced for £20 million bids.
And it adds that the former England number one could be available for a fraction of the £3.5 million Burnley paid Manchester City a year ago, having fallen behind Tom Heaton and Nick Pope in the Clarets' pecking order.
Hart is said to be Blackburn's top goalkeeping target and a move to the Championship could appeal, according to The Sun.
But of the two clubs, it is Stoke who may be in a stronger position financially, as they welcome a second season of Premier League parachute payments.
Both Stoke and Blackburn are also being linked with a move for the former Potters midfielder, Glenn Whelan, who was recently released by Aston Villa.
Would Hart be a good addition for Stoke?
Have something to tell us about this article?