The 33-year-old Cameroon is expected to sign a two-year contract with a news conference expected for Wednesday morning to announce the signing.
Eto'o arrived at the club's Finch Farm training ground just after 2pm to undergo a medical and the deal may be confirmed as early as Tuesday evening.
QPR and near neighbours Liverpool were also thought to be interested in Eto’o, who could complete the move in time for Saturday’s Premier League clash with former side Chelsea at Goodison Park.
He scored 12 goals in 35 games for the Blues last season, but the four-time African Footballer of the Year fell out with manager Jose Mourinho and was released in July.
The introduction of Eto’o will give Roberto Martinez a much needed attacking option as he is still waiting for Arouna Kone to recover from a knee injury.
He joins Romelu Lukaku, another former Chelsea man who completed a £28m move from Stamford Bridge last month, while the west London side have also loaned Christian Atsu to Everton.