Before the start of the campaign, Villa were tipped as one of the favourites to win the second-tier title, with Bruce putting together a strong squad which now includes John Terry after his departure from Chelsea.
However, back-to-back defeats to Cardiff City and Reading have left Villa with just one point from three games, and according to the Daily Star, Bruce cannot afford a similar return over the upcoming period.
It has been suggested that the negative fan reaction after their setback at the Madejski Stadium has been noted by the club's board, with a decision potentially having to be made should Villa not improve during their remaining games in August.
On Saturday, Villa host Norwich City before they make the trip to Bristol City after their League Cup showdown with Wigan Athletic.
Since his arrival last October, Bruce has won 16 and lost 16 of his 40 matches in charge.