Eriksen has been linked with a move away from north London for some time after failing to negotiate a new contract with Spurs.
And Real, who have already signed Eintracht Frankfurt striker Luka Jovic, are expected to make a number of signings this summer and could get Eriksen at a cut price as his current deal expires in 2020.
The Denmark international says he is not aware any concrete talks between Real and Spurs have taken place, but acknowledged it could be time for him to move on,
Speaking to Danish outlet Ekstra Bladet , Eriksen said: "I feel that I am at a place in my career where I might want to try something new. I have the deepest respect for everything that has happened at Tottenham but I have also said that I would like to try something new.
"I hope there will be clarification during the summer. That's the plan. In football you do not know when clarification will come. It can happen anytime. It would be best for everyone if it happens as soon as possible, but in football things take time.
"It depends on Daniel Levy. And another club has to come in. Or I have to sit down at the table and negotiate a new contract. You can't set a date yourself.
"If I have to go, then hopefully it will be a step up. It's [a potential move to Real Madrid] a step up. But that requires Real Madrid to call Tottenham and say they want me. And they have not done so yet, as far as I know.
"It is hard. It depends on the possibilities. If nothing pops up that is more exciting, why not stay at Tottenham? If I am to sign a new contract it depends on the conditions."
Should Eriksen leave Spurs this summer, he would bring a six-year spell at the club to a close having signed from Ajax in 2013.
Spurs chairman Daniel Levy has done a number of big-money deals with the Liga club in the past, negotiating huge fees for Luka Modric in 2012 and Gareth Bale in 2013.