According to Spanish media outlet Sport, both clubs have registered their interest with Barca.
The 29-year-old stopper has a £52 million release clause in his contract and Barcelona won’t entertain any bids below that figure.
Barcelona signed Cillessen from Ajax in 2016, but he recently suggested he was unhappy with his lack of playing time.
“I love this club, but I am a professional and I want to play more than ten games in a season,” Cillessen told Ziggo Sport.
“The management already knows that I want to play more, but I also read that they do not want me to leave and that they are happy with my work.
“I am not the type of player who goes to war with the clubs, I have been here for two years, and, for the moment, I will continue.”
Cillessen, who was recently linked with a move to Liverpool, has made just 21 appearances in all competitions for Barcelona.
He is the current first choice for Netherlands and has been capped 39 times by his country.