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Pepe, 34, has spent 10 years with the European champions, but has fallen down the pecking order under Zinedine Zidane this season as Madrid won a La Liga and Champions League double for the first time in 59 years.
“What Madrid have done is spectacular, but there are things that I still don’t understand,” the Brazilian-born centre-back told Spanish radio station Cope.
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“I don’t know why I have disappeared from the team.
“An era has come to an end and, from now, a new one will start, but Real Madrid’s way of doing things hasn’t been correct.”
French giants Paris Saint-Germain and Italy’s Inter Milan have both been rumoured as Pepe’s possible next destination.
However, he said England also remained a possibility having previously snubbed the option of a lucrative move to China to help his chances of playing for Portugal in next year’s World Cup.
“There are more offers. Not just from PSG or Inter, but England for example,” he added.
“I don’t have anything signed with any other club. I could have done so earlier but I didn’t want to destabilise the team.”
Pepe has won three La Liga titles and three Champions Leagues in his decade in Spain.
However, his career at Madrid was overshadowed by a shocking incident in 2009 when he kicked Getafe’s Javier Casquero twice as he lay prone on the ground, for which Pepe was handed a 10-game ban.
