The 28-year-old Germany international, who has been at Dortmund since 2012, has recently made his comeback from the cruciate knee ligament injury he suffered in the 2017 DFB-Pokal final last summer.
It was the latest injury issue to hit Reus during his time at the Westfalenstadion although he remains a key part of head coach Peter Stoger's plans.
However, with 18 months remaining on his current deal and no sign of an extension being signed, Spurs are believed to be keeping tabs on Reus, with a view to a swoop.
If Reus has still not agreed a contract by the end of the season, Dortmund could be tempted to sell to avoid losing him on free transfer the following year.
Despite signing Lucas Moura from Paris Saint-Germain, Spurs chief Mauricio Pochettino is keen to add more attacking quality to his ranks and a move for Reus could be on the cards.