The Liverpool striker told Uruguayan radio station Sport 890 that his treatment by the British media has affected his family and could force him to continue his career elsewhere.
He said he had talked to Brendan Rodgers, manager of the English Premier League club, and the Northern Irishman knew what he wanted.
The 26 year old has courted controversy since his arrival from Ajax, facing repeated accusations of diving, and serving a ban for racially abusing Manchester United's Patrice Evra and another another for biting Chelsea's Branislav Ivanovic.
Suarez said: "I'm happy at Liverpool, I'm happy because of the fans. I made a mistake, I'm human, but they've talked about me in ways they shouldn't have.
"My family have suffered and things got out of hand. My daughter and my wife have suffered. I'm not prepared to continue to put up with the English press.
"I love Liverpool, but if there is a chance of playing somewhere else...I suffered too much as a kid to get where I am to be attacked unfairly by the English press. They haven't appreciated me as a player, they've just judged my attitude."
Asked about Madrid, he said: "I've talked to my agent, there's nothing concrete. To say no to Madrid would always be difficult and more so with the team they have," Suarez said.
"I have a contract with Liverpool, but it would be very difficult to say no to Real Madrid. There are a lot of rumours in the transfer period, but you never know until firm offers are made."
He added: "I've talked to the coach [Rodgers], and he knows what I want. He was a player. Some of the players understand my situation - enough said."
The striker has established himself as one of Europe's leading players since moving to Anfield in 2011 but he will miss the first six games of next season following his 10-match ban for biting Ivanovic.
Monaco close in on Falcao deal
Monaco are expected to make official their move for Radamel Falcao, Atletico Madrid's Colombian striker, at an end-of-season celebration on Friday evening, according to sources close to the principality club.
The party, which will be held in a marquee near to Monaco's Stade Louis II, will be attended by Prince Albert and the club's billionaire Russian owner Dimitri Rybolovlev.
With Atletico playing their final league game of the season away to Zaragoza on Friday, Falcao is not expected to be present, but it is believed that Rybolovlev will take the chance to announce the striker's signing.
Falcao arrived in the principality on Monday evening and spent Tuesday there with his agent Jorge Mendes, finalising the details of his contract and passing a medical.
He has since returned to the Spanish capital.
Different sources close to the deal say that Falcao is set to sign a five-year deal with Monaco but that a clause will allow him to leave for a major club in June 2014.
Monaco and Atletico have already reached agreement on a transfer fee.
While the player's release clause currently stands at €60 million (Dh285m), reports in the Spanish sporting press indicate that the fee will be €45m, with Falcao earning a net salary of €14m per year.
Neymar to arrive in Barcelona on Monday
Neymar, the Santos and Brazil star, who has just signed a five-year contract with Spanish giants Barcelona, will be presented by his new club on Monday, Brazil's technical director Carlos Alberto Parreira said on Wednesday.
Speaking at a news conference, Parreira confirmed that Neymar had been given permission to head to Barcelona after playing for Brazil in Sunday's friendly match against England in Rio de Janeiro.
The 21 year old, whose full name is Neymar da Silva Santos Junior, revealed on Tuesday that he was keen to join Barcelona but wanted first "to focus on the Confederations Cup to try to win the title for Brazil".
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