The Dutch side's sporting director, Marc Overmars, confirmed the club are in negotiations about signing the former Chelsea striker on a short-term contract
Eto'o and his agent were reported to have met club officials in Paris on Friday to thrash out a deal after the striker was released by Chelsea.
The Cameroon international was believed to have wanted to stay in England and had been in talks with a number of Premier League clubs including West Ham, QPR and Sunderland.
The Black Cats were leading the race to sign Eto'o after the player's wage demands had put off QPR and the 33-year-old had rejected a move to The Hammers.
However, Overmars has now confirmed that Eto’o is nearing a move to Ajax in the coming weeks.
"Samuel seemed very hungry to play for Ajax. He is fit and has a great drive to succeed once again," Overmars told Voetbal International.
"We don't want to block the growing process of our talents like [Arkadiusz] Milik and [Richairo] Zivkovic, but if it is possible to sign a world class player like Eto'o, then why not? It will be fantastic."
Eto’o scored 12 goals in all competitions for Chelsea last season - all coming at Stamford Bridge - but the Blues opted not to extend his one-year deal, which left the player looking for a new club.