The England international has been a long-term target for the Reds, and Brendan Rodgers reportedly tried to bring Sturridge to Anfield in the summer but an agreement with Chelsea could not be reached.
The report claims Liverpool have already approached Chelsea over Sturridge for the January transfer window, but the Stamford Bridge's stars excessive wage demands could prove a stumbling block.
Sturridge reportedly wants £80,000-a-week and a promise to play through the middle, but the Anfield club are thought to be relectuant to shell out such an amount for an unproven player.
And there is a feeling at Liverpool that Sturridge is yet to justify that kind of outlay. The former-Manchester City star enjoyed a successful start to last season, bagging 10 goals from a wide position under then-Chelsea manager Andre Villas-Boas.
But, under Roberto Di Matteo, and now Rafa Benitez, the England forward has barely featured - scoring just five goals in an 11-month barren spell.
Despite his mixed goalscoring record, the 23-year-old is believed to be insisting he plays through to middle should he move to the Reds next month.
Liverpool have just one fit senior striker - Luis Suarez - and have been linked with reinforcements in January. But Chelsea are also short of options up front, Rafa Benitez can only call on Fernando Torres and Sturridge at the moment.