The Saints have already sold captain Adam Lallana and star striker Rickie Lambert to the Reds, while emerging left-back Luke Shaw is now at Manchester United.
Dejan Lovren and Morgan Schneiderlin are also being linked with switches away from Hampshire, which could leave Ronald Koeman with a massive job on his hands before the season starts.
Rodriguez was expected to be a big player that would be staying put, but now it has emerged that contract talks with the attacker have stalled.
The England international – who is currently out of action with a knee injury – has been discussing an extension, yet with Liverpool and Spurs circling he is keeping his options open.
Both clubs were impressed with the 24-year-old last season, and could swoop if given encouragement by either the player or Southampton.
Rodriguez is understood to be willing to wait until the end of the transfer window to make a final decision, with the identities of incoming players a big factor for him.