According to the Herald Sun, Bruce will likely receive a lucrative three or four-year offer from the Bulldogs.
St Kilda are said to be aware of the interest in Bruce, who kicked 36 goals for the club in 2019.
The Bulldogs move for Bruce is said to be driven by former Saints list manager Ameet Bains, who now works as CEO of the Bulldogs.
Bruce is contracted for one more season with St Kilda, however the Saints are yet to make significant moves to secure his long-term future.
The 27-year-old placed fourth in the league for total contested marks and would be an exciting pairing with young Bulldogs forward Aaron Naughton.
Facing a rebuild after eight years of missing out on finals, St Kilda are likely to want at least a second round pick for their 197 centimetre forward.
Earlier reports also suggest the Saints may also be moving on from key tall Jake Carlisle, opening up space for young tall Max King to gain experience in 2020.
The Western Bulldogs are also expected to be active in the trade period, with salary cap space being made available after missing out on Chad Wingard last year.
The Bulldogs head into finals as one of the most in-form teams in the competition and will go up against GWS in Sydney on Saturday afternoon.