According to the Mirror, Real have made Sterling their number one transfer target after becoming encouraged by the simple fact that the England international has yet to be offered a new long-term deal with financial terms that far exceed his current £30,000-per-week wages.
Indeed, further speculation on the matter from the Daily Star suggests that Real - who had endured somewhat of a tepid start to the new domestic season prior to their 8-2 demolition of newly-promoted Deportivo on Saturday - will use the £59.7 million generated from the summer sale of Angel di Maria to Manchester United in order to fund a move for Sterling.
Liverpool will presumably be desperate not to relinquish the services of a player who has consistently proved himself to be one of the Premier League's most exciting talents and also become a central figure on the international stage under Roy Hodgson.
The 19-year-old - who first moved to Merseyside from Queens Park Rangers in 2010 - scored nine goals in 33 appearances last season and was an integral member of the Liverpool attack that so nearly spearheaded the club to their first top-flight title in over two decades.
So far this season Sterling has been in similarly imperious form, although his excellent strike in Saturday's clash against West Ham was not enough to avoid a disappointing 3-1 defeat to the Hammers at Upton Park.
Give that he is arguably now Liverpool's star player - an honour that could also be claimed by injured striker Daniel Sturridge - following the high-profile sale of the controversial Luis Suarez to Barcelona this summer, one would have thought the club would do everything possible in their power to ensure that Sterling's long-term future was secured and guaranteed as soon as humanly possible.
While a new contract offer for Sterling is surely forthcoming, it will be intriguing to see if the player himself retains any interest whatsoever in a potentially lucrative move to the Santiago Bernabeu.
Real - who won their 10th European Cup with victory over rivals Atletico in May - are obviously one of the biggest clubs in the world and the opportunity to line-up alongside the likes of James Rodriguez, Cristiano Ronaldo and Gareth Bale whilst earning good money and competing for regular trophies may be too good to resist.
Ultimately, however, you expect to see him remain a Liverpool player for some years to come.