The Bundesliga outfit are keen to sign the 28-year-old on an initial loan deal after a move for Chelsea's Renato Veiga stalled.
When Dortmund first opened talks with Veiga, they were under the impression he would be available on loan.
But Chelsea will only entertain a permanent £20million deal.
This has led to BVB to turn towards Arsenal full-back Zinchenko.
The Ukrainian has fallen down the pecking order under Mikel Arteta and a move to the Bundesliga giants this month is now a possibility.
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Zinchenko has been limited to just seven Premier League appearances this season - starting on just two occasions.
Riccardo Calafiori and Jurrien Timber have been preferred instead of Zinchenko.
And Hale End academy graduate Myles Lewis-Skelly is currently enjoying a spell in the starting XI over Zinchenko having impressed Arteta.
Arsenal paid Manchester City £32million for Zinchenko in 2022 and despite holding down a regular spot during his first campaign, he has since slipped down the ladder under Arteta.
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A move this month would provide the Zinchenko with the perfect opportunity to stake his claim ahead of what could potentially turn into a summer switch.
The left-back could also be buoyed by the loan moves of Jadon Sancho and Ian Maatsen to Dortmund last January.
Both players were hampered by a lack of game time at respective club's Manchester United and Chelsea before playing key roles in helping the Bundesliga side reach the Champions League final.
Their heroics earned them much-needed summer switches from their parent clubs which saw Sancho move to Chelsea, while Maatsen swapped Stamford Bridge for Aston Villa.
Zinchenko has made 79 appearances since arriving at Arsenal, registering two goals and four assists.
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