The German international has yet to pen a new deal and will be available on a free transfer at the end of the season unless fresh terms can be agreed.
As things stand that looks unlikely and Can is clearly waiting to see what offers come in before deciding his next move, and Real's interest is sure to turn his head.
Italian champions Juventus are known to be interested, while there have also been suggestions that Manchester City could offer him a bumper deal at the Etihad Stadium.
Now Real have also entered the race and the 23-year-old will be keen to hear what Zinedine Zidane has to offer.
Liverpool's resolve to keep hold could be tested in January with reports claiming offers in the region of £20million could be made and they may be tempted to cash in.