Pepe, 34, is out of contract at the end of the month and wanted a two-year deal at Madrid.
But, after being offered just a one-year extension, he has decided to move on from the club he joined from Porto in 2007.
"It is clear that I will not continue at Real Madrid," he told Cope.
"One stage is over and, from now on, another one will start."
"There are more offers, also from England," Pepe said.
Zinedine Zidane's Madrid enjoyed a spectacular campaign that included winning La Liga and the Champions League.
But Pepe has plenty of questions over his exit, saying: "I don't want to blame anyone. What Zidane has done with Real Madrid is spectacular, but there are things I don't understand.
"I didn't say goodbye to Zidane because Zidane and Real Madrid knew I was going before I did."