The Reds are thought to have made the Monaco forward their top target, but interest from Juventus and Real Madrid is understood to be strong.
A loan move for the Colombian has been mooted, but a fee just for the season would likely be close to £10m, which would offer Liverpool no guarantees beyond the 2014/15 campaign.
With Brendan Rodgers desperate to get in a new forward to support Daniel Sturridge and Rickie Lambert, Eto’o is once again on the agenda.
The Merseysiders have reportedly spoken to the player’s representatives already, but were deterred by his £100,000-per-week wage demands.
They offered around half of that figure, yet despite Eto’o’s desire to stay in England he opted against penning the contract.
Spurs and bitter rivals Everton are also keen, but neither can present him with Champions League football and may bow to his pay wishes.