The Paraguayan international was a free agent after leaving Wigan Athletic at the end of last season, but linked up with his former boss at the Latics after penning a two-year contract with the Toffees.
"We talked to a lot of clubs but I was always interested in staying in the Premier League," Alcaraz told PA.
"I like this competition and at this moment in my career I thank Everton for giving me this opportunity to continue in the Premier League.
"I know him [Martinez] really well and he knows me very well also. We had a very good relationship in the three years at Wigan and I want to continue that. Nothing is going to change, he is very professional."