"If you can get a player with his qualities and he still has ambition to show what he can do, then it is an opportunity not to be missed," De Boer told Fox Sport television on Sunday ahead of his team's 4-1 win over Vitesse Arnhem in their opening game of the new Dutch season.
De Boer said he had received the suggestion from the club's technical director Marc Overmars via an SMS that said: "Eto'o".
"I texted him back: 'If you can get him, that would be top. As far as I'm concerned you can try'. He (Eto'o) has been in the game for 20 years so he can naturally take a leadership role. He can teach our own players about professional football," De Boer added.
The 33-year-old Eto'o held talks with Overmars in Paris on Thursday.
The four-times African Footballer of the Year was released by Chelsea at the end of June after one season and his World Cup appearances were limited in Brazil because of a knee injury.
Ajax's rivals PSV Eindhoven said they had also been offered the services of Eto'o but the striker's wage demands proved to be the sticking point.
"He would like to earn double than our best paid player... and that we can't agree to," PSV director Marcel Brands told Fox.
In 2011, Eto'o signed for Russian club Anzhi Makhachkala in a deal that earned him 20 million euros ($26.82 million) a year.
Eto's has won Champions League titles with Barcelona and Inter Milan.