There were suggestions that the pair, who were also together at Inter Milan, had fallen out when Chelsea manager Mourinho made a joke about Eto'o's age.
But the four-time African Player of the Year told a news conference on Wednesday: "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."
Everton manager Roberto Martinez suggested Eto'o may not be ready for Saturday's match at Goodison Park but the Cameroon international said: "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.
"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."
Martinez added: "In every aspect he is the perfect fit for the team and squad. His mental focus was so impressive last season I knew for a fact that he was exactly what we needed for the squad.
"I would have liked him to have 10 or 11 sessions before putting him in a competitive environment, but I quickly learned I was wasting my time.
"He wants to be helpful to the team and be involved."
Eto'o's debut for Chelsea last September was in the 1-0 defeat at Everton and he went on to score nine league goals in 21 appearances for the west London club.