The striker is keen to stay in the Premier League after leaving Chelsea, with Everton and West Ham also looking into a deal for the 33-year-old
Samuel Eto'o will hold a second round of talks with Liverpool later this week after failing to agree a deal during initial discussions, Goal understands.
The Cameroon international wants a one-year contract worth a basic £135,000 a week, while Liverpool have offered a lower package but are willing to top up the 33-year-old's wages with performance-related add-ons.
Discussions between representatives of the striker and Anfield officials took place on Tuesday and further talks are planned for Thursday as the Merseysiders seek proven back-up to Daniel Sturridge and Rickie Lambert.
Ajax, Everton and West Ham are among the clubs keen on negotiating a deal for the veteran, while there is also strong interest from a number of Italian clubs.
The Cameroonian was released by Chelsea at the end of last season and is currently a free agent.
Liverpool have been scuppered in their attempts to sign a direct replacement for Luis Suarez despite selling the Uruguayan for a club-record fee of over £70 million.
Loic Remy failed a medical after agreeing a £9m switch from QPR and other negotiations are yet to bear fruit due to price and availability.
However, manager Brendan Rodgers continued his extensive summer squad overhaul on Tuesday as Liverpool agreed a deal with Sevilla for left-back Alberto Moreno worth an initial £12m rising to £16m.