Spurs were strongly linked with signing the striker in the summer, but with the forward recovering from a serious knee injury, any move was put on ice.
Rodriguez enjoyed a terrific season last year at Southampton, under new Spurs head coach Mauricio Pochettino, scoring 17 goals.
A potential deal for the attacker would cost upwards of £15 million, potentially more if he were to recapture that same form.
The Telegraph claim that Spurs are so determined to sign Rodriguez, that they decided not to attempt to sign Danny Welbeck on transfer deadline day, leaving Arsenal with a clear path to snap him up from Manchester United.
"Tottenham were interested in signing Danny Welbeck on loan from Manchester United in August, but did not make a permanent bid because of Pochettino’s desire to try again for Rodriguez."
This suggests the club held back funds for a potential Rodriguez deal, and showing how serious Pochettino is about re-signing a player he clearly rates highly, certainly higher than Danny Welbeck.
Rodriguez is currently aiming to return to action in the middle of November, giving Spurs time to assess his fitness before the January transfer window.
Southampton's sensational form this season throws a potential spanner in the works however. Rodriguez could become convinced to stay if Tottenham slip further behind the Saints.
If he does push for a move, Southampton fans won't worry bearing in mind the way they have bounced back from selling other 'key players' during the summer to re-invest the funds and come out an improved side.