The Reds have offered the Cameroon striker a one-year contract but the Toffees are hopeful that the promise of a further season in English football will make the difference.
Liverpool, which is set to confirm the arrival of Mario Balotelli, is also looking to bring in Eto'o as Brendan Rodgers looks to bolster his squad ahead of a return to the Champions League.
The Reds have offered the Cameroon international a one-year deal, but he is tempted by the security of Everton's offer of a further season in English football.
The 33-year-old has been linked with a number of clubs this summer after leaving Chelsea in July, but would prefer to remain in England after enjoying his time at Stamford Bridge.
He scored nine goals in 21 Premier League appearances last season, but was embroiled in a row with Blues boss Jose Mourinho after he joked that he did not know the striker's real age.
Eto'o branded the Portuguese a "fool" and a "puppet" towards the end of his time in west London, and insisted he can still cut it at the top level.
Everton smashed its transfer record to sign Romelu Lukaku for 35 million euros from Chelsea earlier this summer, and is now looking to add depth to its striking options by bringing in Eto'o.
Muhamed Besic, Gareth Barry and Christian Atsu have also been brought to Goodison Park this summer.