Real are thought to demand an asking price in excess of £20m for the Spanish youngster, a price tag that could turn Liverpool away.
Liverpool have faced competition in the past for Rodriguez, with North London sides Arsenal and Tottenham Hotspur having eyed up the 21-year-old in times gone by. Further afield, Italian giants Juventus have considered a move for him.
Jese has made 14 appearances for Real in the league this season, but has failed to complete a full 90 minutes. He has two league goals and assists in La Liga.
The Spaniard has made four appearances in the Champions League this campaign. He scored his first-ever goal in the competition last week, making a swift impact from the substitutes bench when he doubled Real’s lead against Roma after Cristiano Ronaldo had opened the scoring.