The 27-year-old has played a crucial role in moving Hull second in the Championship, three points clear of third-placed Watford with five games to play.
Since signing on loan for a second time in January, he has conceded just 19 goals in 19 games and notched up eight clean sheets.
Stockdale remains behind Mark Schwarzer in the pecking order at Craven Cottage and Bruce revealed early discussions have already taken place to keep him at the KC Stadium.
He told the Hull Daily Mail: "I don't think it is set in stone, but it is something that has been discussed.
"There is certainly the possibility of a deal to be done if we were to get to the Premier League. That is where Stockdale wants to play and that is where we all want to play.
"He has played a big role from the minute he walked in through the door here. And he has still got a big part. He handles the situation well and that is where his experience comes in.
"It is important to have that position as a position of strength and that is what he gives us. You can count on one hand the number of mistakes he has made, so he has been very influential."