The 24-year-old, often utilised at right wing-back, featured 40 times for the Austrian champions across all competitions during the 2021-22 campaign, racking up an impressive return of eight goals and three assists in the process.
Bees boss Thomas Frank is said to be targeting both wing-back areas this summer, and according to The Sun, Kristensen is the primary target for the right-hand side as things stand.
The report suggests that a £10m transfer fee would be enough to secure his signature, with Frank hopeful of using his connections to bring his fellow Dane to West London.
However, Leeds United are also said to be interested in making a move in the coming weeks, with Jesse Marsch keen to be reunited with Kristensen after taking him to Austria from Ajax for just over £4m in 2019.
Since making that move, the FC Midtjylland academy product has gone on to make 72 appearances for Salzburg across all competitions during his three-year stay so far.