The Reds have emerged as shock contenders in the race for the 33-year-old Cameroonian over the course of the past few days, with the forward desperate to stay in England after one season at Chelsea.
Liverpool have this summer lost Luis Suarez, leaving Brendan Rodgers with only Daniel Sturridge and Rickie Lambert as orthodox central striking options as Fabio Borini nears a switch to Sunderland.
Initial discussions between the player and the club revealed that Eto’o wanted £135,000-per-week, which the Merseysiders were far from keen to match.
But further talks are scheduled, with the Anfield outfit reportedly willing to get closer to his demands with performance related add-ons.
Local rivals Everton also want the veteran African star, while West Ham are still keen despite a breakdown in negotiations with his representatives a few weeks ago.
Liverpool attempted to sign Loic Remy from QPR recently, but problems discovered in his medical saw the deal collapse.