The Argentine goalkeeper has not made a single competitive appearance for Jose Mourinho’s side this season, featuring on the bench just twice so far.
Gazzaniga has fallen down the pecking order at Tottenham, with summer arrival Joe Hart taking over as club captain Hugo Lloris’ deputy.
With playing time hard to come by at Spurs, there have been suggestions that the 29-year-old could part ways with the north London outfit during the ongoing transfer window.
It was claimed that Gazzaniga is looking for a new club in Spain and La Liga side Elche have emerged as a potential destination for the goalkeeper.
Elche currently sit 19th in La Liga and are in the middle of a fight to retain their top flight status.
The Spanish side will be hoping that Gazzaniga can help them move away from the relegation zone in the second half of the season should they sign him from Spurs.
With time running out in the transfer window, it remains to be seen if Elche can strike a deal with Tottenham for the Argentine.