The 33-year-old forward is currently a free agent after being released by Chelsea, and despite serious, as well as lucrative, interest from the Middle East, he wants to stay in England.
Liverpool were considered to be frontrunners for his services, but with the Reds having pulled out of the running to bag Mario Balotelli, their bitter cross-city rivals have seized the momentum.
Roberto Martinez’s side are understood to have offered the Cameroonian a contract, but the details are as of yet unknown.
There has been talk of Eto’o chasing a £100,000-per-week salary, but Everton are likely to be looking at a lower wage with the security of a two-year running time.
The Merseysiders are known to be looking to bulk out their squad to cope with Europa League football, with £28m record signing Romelu Lukaku short on back-up.
Danny Welbeck of Manchester United is also a target.