The Uruguay international, currently in South America for the Confederations Cup, had earlier stated that he would like to link up with Cristiano Ronaldo in the Spanish capital.
And in a second interview, the 26-year-old has reiterated his desire to leave the Premier League while refusing to rule out a move to Madrid.
"Any player would like to play for a great team and Real Madrid certainly are that," Suarez told television show RR Gol.
|4/7||Luis Suarez is 4/7 with Paddy Power to join Real Madrid
"Obviously Real Madrid is a great team and every player dreams about them; Real Madrid is Real Madrid and it's as high as you can go as a player and they will always tempt you.
"I'm not sure if I see myself in the white [Madrid], the only thing I know is that I have a contract."
Suarez, who is currently serving a domestic suspension for biting Chelsea defender Branislav Ivanovic, is adamant he wants to leave Anfield because of his harsh treatment by the British press.
"Ever since I arrived I have felt bad, they have never judged me for my play but with the attitude that I dive, protest, makes gesture, racism... everything.
"They have never spoken well of me. For that reason, when I was told I could win the Player of the Year award, I said: 'I will not get excited as I know that I will never win an important award in England.'"
The 26-year-old also slammed the length of his racism ban compared to that of John Terry, who received half the punishment for an incident involving Anton Ferdinand, and believes he can never forgive or forget his harsh treatment despite his love for Liverpool.
"Without any proof they gave me an eight-match ban, but with Terry, where they had proof, lip-readers, they gave him four, I'm South American and I think that's the root of all of this," Suarez continued.
"Liverpool is a club that I dreamt about playing for as a boy. I have been with great players there, they know me and I would love to continue for many more years at Liverpool. The stadium is spectacular, the fans are phenomenal: people with tattoos of me, things that I could never have imagined.
"After the [Evra] suspension people would come up to me, this is something that really touched me and they would say: "Luis forget everything and stay."
"But I can't forget as it was something that I had to suffer, that my family had to suffer and away from football I have a family to support and who are hurt by these things."
Suarez went on to describe one such example of his family being subjected to abuse, describing a shopping trip that almost brought his family to tears.
"About a week ago I was walking in a shopping centre near Manchester and three or four guys asked me for a photo," he added.
"While we were posing for it my wife said to me "Luis, get out of the photo". She noticed they were making biting gestures. I was with my wife and my daughter. Things like that get on your nerves. My wife was on the brink of tears and the blokes ran off laughing. You get tired of stuff like that."