A statement from Shaw was released on Blackburn's official website, on Wednesday night confirming their interest in the 37-year-old former Portsmouth boss, with a Blackpool statement acknowledging a "formal approach" shortly afterwards.
Despite no further official line from Blackpool it was widely reported on Wednesday morning that Rovers' bid to secure Appleton as Henning Berg's successor at Ewood Park had been knocked back.
Appleton left Portsmouth to take the reins at Blackpool on November 7 following Ian Holloway's departure to Crystal Palace, seven days on from Berg's own appointment at Rovers.
But a run of one win in 10 games, including a 2-0 defeat to Appleton's side at Bloomfield Road in his penultimate match, saw the ex-Norway international sacked after 57 days in charge.
Rovers knocked back Alan Shearer, who applied for the job, and - once Berg was sacked - Shearer was quick to distance himself from the job, saying: "I'm not interested this time."
Could this be another Blackburn disaster?
Blackburn have won all three games since Berg was dismissed and last week issued a statement to confirm reserve team manager Gary Bowyer and head of youth Terry McPhillips would remain in caretaker charge until the end of this month.
This would now appear unlikely in light of the Appleton approach.
Appleton has won only two of his 11 Championship games in charge of Blackpool.