The 2018 summer signing is training with old club Middlesbrough.
Gibson has only played one game for Burnley this season. He cost the club £15 million, but is said to have suffered a breakdown in relations with the manager.
However, boss Dyche does not want to make his communications with Gibson public, as he is choosing to protect the defender.
Dyche told reporters: "It's quite a unique situation.
"There's a private angle to it, which will remain private, but at this time he's with Middlesbrough, training there to keep fit and stay sharp.
"He's still a Burnley player, though."