De Gea, 24, is expected to join Real Madrid this summer and was dropped by manager Van Gaal for their Premier League opener with Spurs last weekend.
And Cillessen, along with Galatasaray ace Fernando Muslera, has reportedly been lined up as a replacement for the Spain international.
However, 26-year-old Cillessen spoke out over those claims, insisting he has not discussed a move with former Netherlands coach Van Gaal.
"I've read that this weekend I have spoken with Van Gaal, but the last time I did was during the Sports Gala late last year," Cillessen told Ajax Show Time.
"Also my management this weekend has not heard from United."
Jasper Cillessen - Spotlight on
Invalid email
We use your sign-up to provide content in ways you've consented to and to improve our understanding of you. This may include adverts from us and 3rd parties based on our understanding. You can unsubscribe at any time. Read our Privacy Policy
Meanwhile, Van Gaal is expected to secure the arrivals of two more outfield players this summer to push his overall spent at Old Trafford close to £300m.