The 22-year-old Spaniard only joined Los Blancos last year, but is already thought to be considering a move away from the Bernabeu after a difficult first season.
Isco managed 32 La Liga outings in all competitions during his debut campaign in Madrid, and with Toni Kroos having been snapped up from Bayern Munich his chances of building his role appear to be bleak.
Real are understood to be ready to listen to offers, which has alerted Liverpool.
The Reds have cash to spend following the sale of Luis Suarez, and are known to be keen to get top class players to Anfield to ensure that they can build upon 2013/14’s impressive second place finish.
Manchester City are also in the running, but Manuel Pellegrini is thought to be more keen on a loan move for the player he coached while at Malaga, which may put the Spanish club off as they look to raise funds for moves for James Rodriguez and Radamel Falcao.
Isco is able to play in any role behind a main striker, and could be a player that would fit well with Brendan Rodgers’ playing style.