The Brazilian international – who was strongly linked with Arsenal over the summer – only joined the Bundesliga side from their domestic rivals Bayern Munich a matter of months ago, but that has not stopped Spurs from taking an interest.
Andre Villas-Boas is keen to dip into the transfer market once again in the New Year, with his £110m spend during the last window having failed to garner the desired effect.
Paulinho and Etienne Capoue were both brought in as a part of the overhaul, but the Portuguese coach is thought to be concerned at the lack of dynamism currently being shown by his central players.
Gustavo is believed to be somebody that AVB feels can inject some energy into his ranks, which could bring about a release clause-triggering offer.
It is believed that a figure in the region of £20m will be enough to grant Tottenham permission to talk to the 26-year-old, who may be open to making a switch to the Premier League.
The North London club have also been linked with Basel’s Mohamed Salah, who is attracting interest from Manchester United and Liverpool.