Spanish publication Sport (via Sport Witness) suggest that the Dutch stopper is prepared to submit a transfer request should the offer from Liverpool prove to be genuine, while the other club said to be interested has not been named by the newspaper.
The goalkeeper is said to have a €60m release clause in his contract and Barcelona have previously been reported as unwilling to sell for any less than that. However, it is possible that this could change should Cillessen demand a move away from the club this summer.
Cillessen has spent the last two years at Barcelona but has spent most of that time serving as backup to first-choice goalkeeper Marc-Andre ter Stegen, featuring almost exclusively in cup matches for the Catalan side. This could make the prospect of a move to Anfield more tempting, with Cillessen given the chance to claim the number 1 jersey at Liverpool.
Liverpool manager Jurgen Klopp has been vocal in his support of current goalkeeper Loris Karius over the past week, with the German still the subject of criticism and ridicule following his disastrous performance in the Champions League final.
Karius was later revealed to have been playing with a concussion for much of the match, which could account for his glaring errors which led to two of Real Madrid's three goals.
And while Klopp has suggested he will continue to place his trust in Karius next season, Cillessen could view the place in Liverpool's starting XI as being there for the taking given the German's track record of making high-profile errors in matches.
Cillessen made a name for himself at Dutch side Ajax, making 143 appearances in five years at the Amsterdam club. The goalkeeper has also been capped 40 times by the Netherlands, featuring prominently at the 2014 World Cup in Brazil.