Eto'o labelled the Chelsea boss as a "fool" and a "puppet" after the Portuguese made a comment questioning the striker's age during his spell at the London club last season.
The 33-year-old former Real Madrid and Barcelona forward joined Everton on a two-year deal after much speculation over his future and could make his debut against Chelsea on Saturday.
Eto'o was in no mood to start another row with Mourinho at a press conference on Wednesday, though, as he expressed his gratitude to his former manager.
He said: "The only thing I have to say about Jose Mourinho is to thank him.
"It's down to him that I had the opportunity to be accepted and integrated into the Premier League and I want to thank him for that.
"I'm not particularly concerned that it's against my [old] team [on Saturday].
"The important thing is to integrate into my new team-mates' way of playing and get ready for that match. If I don't feature it's just important to get ready."
Eto'o revealed that he had many options after leaving Stamford Bridge, but Everton boss Roberto Martinez made the biggest impression on him.
"In the last 10 days I've travelled quite a lot and spoken with quite a few managers. I'm nearer the end of my career and I wanted to fit in where my contribution will be valued." said Eto'o.
"Over the course of those travels the most inspiring and the person that said my contribution will be valued was Roberto."
