The 29-year-old has played second fiddle to Marc-André ter Stegen since he joined the Catalan giants in 2016, but the Dutch shot stopper is now convinced he has to leave the club to realise his desire of becoming first choice at a club.
According to Mundo Deportivo, via Sport Witness, Liverpool have made contact with Cillessen to offer him the opportunity to become the number one at Anfield next season - a proposal he is understood to be seriously weighing up.
With Loris Karius and Simon Mignolet under performing and Danny Ward yet to show off his credentials, Cillessen would be all but guaranteed a run as the first choice keeper if he were to move to Merseyside.
However, Cillessen is aware of the potential for a goalkeeper domino effect to sweep across Europe which could see Thibaut Courtois leave Chelsea for Real Madrid, leaving the 29-year-old to bide his team to see if he would emerge as a target for the Blues who are under new management.
Cillessen is said to be content about the possibility of continuing with the Spanish giants, but a host of Premier League clubs are understood to have made contact with Barcelona to inquire about the 29-year-old's availability - effectively leaving the ball in his court.
The Reds have been linked to a host of shot stoppers this summer, but a deal for their main transfer target in Roma's Alisson has proved difficult to navigate due to the Italian club's demands.