Per Lluis Miguelsanz of Sport, the Netherlands international has been strongly linked with the Reds, who are expected to prioritise the position this summer. They're not the only club looking for a goalkeeper, however.
According to the report, Chelsea could be tempted into selling Thibaut Courtois, as the Belgium international has no intention of renewing his contract. The Gunners' top option in goal is veteran Petr Cech, but the 36-year-old isn't the same player he was a few years ago.
Liverpool have Loris Karius and Simon Mignolet, but the latter could leave the club this summer and the former blundered in the UEFA Champions League final. His mistakes cost the Reds in the 3-1 loss against Real Madrid, and at the very least, they'll want to add some competition.
Cillessen has only been at the Camp Nou since 2016, but he is a backup behind Marc-Andre ter Stegen and at the stage of his career where he needs to play regularly.
Per the report, Barcelona have no intention of selling the 29-year-old for a fee below his buyout clause, which stands at €60 million (£52 million).
Bleacher Report's Sam Tighe thinks that is way too much:
ESPN FC's Samuel Marsden is also a fan of the fit:
The report also states a lot will depend on Cillessen, and one would have to assume he will push for a move if the right offer arrives.
The former Ajax man is a solid goalkeeper with good reflexes and the ability to be involved in the passing game. He's not elite and doesn't have much room left to grow, which may scare off a few suitors or keep them from spending a good deal of money.
With transfer fees continually rising, though, Cillessen could be a value signing if a club is able to talk Barcelona down to a fee in the region of £30 million.