The 24-year-old now plies his trade in his homeland with Valencia, and has been in good form of late after starting the season as a backup player at the Mestalla.
Rodgers has noticed his recent rise, and is now considering offering him a second chance to impress in England.
The silky passer spent time on loan from Real Madrid at Loftus Road earlier in his career, but failed to establish himself with the Super Hoops.
Since returning to Spain, initially with Getafe, he has rediscovered the form that once saw him hailed as one of La Roja’s rising stars.
Rodgers sees him as a good potential addition to his squad as he plans for the possibility of European football, with the Reds’ squad relatively short on depth as it stands.
Parejo can operate as an attacking or defensive midfielder and has been known to play from a wide position, which is the sort of versatility the Northern Irishman is chasing.
Valencia paid Getafe £5m for Parejo’s signature in 2011, and may look to make a significant profit if they are to sell.