The 20-year-old striker's burgeoning reputation comes off the back of a successful tournament with Spain at the U21 European Championships in Israel, where Morata picked up the Golden Boot award en route to winning the trophy.
Reds boss Brendan Rodgers is part of a growing list of Premier League managers said to be monitoring the youngster's ongoing contract situation at the Santiago Bernabeu, and is weighing up a bid to bring him to Anfield ahead of the 2013/14 campaign.
With less than two years to run on his existing deal at Real Madrid, Morata could look to negotiate a move away from Spain, particularly if he sees a lack of first-team opportunities under new manager Carlo Ancelotti this season.
Liverpool would probably be able to offer the player more regular action, with Rodgers's reputation for bringing through young talent standing the Merseyside club in good stead to secure his signature.
However, Real Madrid remain confident that Morata will snub reported interest in his services and pen new terms with the club, and hope to make an announcement before the end of the week.
With question marks still hanging over the future of Luis Suarez at Liverpool, it's understandable that the club are already on the lookout for striker reinforcements.
If they are serious about signing Morata, they could use the Uruguayan as a makeweight in the deal, given Real Madrid's previous interest in the South American hit-man.