Per Melissa Reddy of Goal, Sevilla have only asked whether Sturridge is available, while Inter have submitted a "serious" proposal for him and offered to pay a significant loan fee, cover his wages and include the possibility of a permanent deal.
Reddy added Sturridge has been encouraged by the professionalism of Inter's approach and their willingness to sign him up.
The 28-year-old has become a fringe player at Anfield under manager Jurgen Klopp.
He is behind Roberto Firmino in the pecking order for the No. 9 role and has been afforded just five Premier League starts this season, the last coming on Nov. 25.
With the 2018 FIFA World Cup on the horizon, Sturridge needs regular first-team action if he is to have any chance of being selected in Gareth Southgate's England squad for the summer tournament.
If he were to move to Inter, there is little chance he would unseat captain and leading goalscorer Mauro Icardi as the first-choice frontman, but Sturridge could theoretically take up a role behind the Argentinian at the San Siro.
He would also be in with a chance of helping the Nerazzurri win silverware, although five games on the bounce without a win has seen them drop from top of the Serie A table to third, nine points back from current leaders Napoli.
Sturridge's career has stalled at Liverpool of late, and the Reds are reportedly open to him leaving in a permanent deal, but only for a £30 million fee, per James Pearce of the Liverpool Echo:
Any move to Inter in January would almost certainly be on loan, but the suggestion the Italians could be open to a permanent move in the future could help them seal the signing of Sturridge before the transfer window closes.