The Uruguay star is widely regarded as one of the best players in English football but has endured a troubled time in the Premier League, showing genius on the pitch but serving eight and 10-match bans for racially abusing and biting opponents respectively.
Liverpool failed to qualify for the Champions League again, so the former Ajax forward has been linked with a number of Europe’s top clubs, including Real.
And with Suarez and many associated with Liverpool believing the press and FA have an agenda against the player, he said he could walk out on England.
“I have spoken with my agent and saying no to Real Madrid would be difficult,” Suarez said in an interview with Spanish radio station COPE’s Latin American correspondent Martin Charquero.
“I am happy with Liverpool, I love the club, I have nothing but praise for the fans, but this year was very difficult for me, my wife and my daughter.
“I have spoken to Brendan Rodgers, and he knows what I want. I have a family, a daughter and I am not prepared to continue this way with the English press.”
He did, however, deny that he was already in talks with managerless Real, as had been reported in the Spanish press.
"So far I have not received any official offer from Madrid," he confirmed, although any contact is likely to come after Real appoint a new manager.
Suarez has also been linked with Atletico Madrid, Juventus, Bayern Munich and Paris Saint-Germain.
In a separate development, Liverpool have been linked with Celta Vigo striker Iago Aspas.
Aspas, 25, has scored 12 goals for the Liga strugglers this season. Celta must win their final game of the season and hope results go their way to avoid relegation.
He would cost in the region of £8 million.