Allardyce was in attendance at Sunderland's development squad's latest match against Borussia Monchengladbach, as he looks to gauge the level of talent available to him at the Stadium of Light.
The Black Cats drew 1-1 with their German opponents, and Stockdale says that he was keen to ensure that the group did not get carried away, with the club's new manager watching them closely.
“We made quite a big deal before the game of not trying to do things that aren’t natural to them,” Stockdale said. “It’s understandable in a way, but we tried to tell them not to try to look good, but to be good."
Allardyce has been tasked with guiding Sunderland to safety, and it is as yet unknown whether he will place his faith in the side's youngsters.
Most of Sunderland's recent managers have been reluctant to give the team's academy graduates regular opportunities in their first-team, with Duncan Watmore the only talent of note to break through.
Throwing more youngsters into Sunderland's team could be a risk for Allardyce as things stand, due to the mounting pressure surrounding the camp.
The Black Cats are still looking for their first win in the Premier League this season, and they need to start hitting form soon if they are to avoid being relegated out of the top tier.