The Sunday Mirror says the Dutch ace, who is not a regular in the new-look Real Madrid line-up, has been given until this Wednesday to make up his mind.
Tottenham and Liverpool have also shown an interest.
But Galatasaray boss Rijkaard believes he can persuade his fellow Dutch countryman to sign a two-and-a-half-year deal worth £8m.
Van Nistelrooy said: "Frank called me and asked 'Do you want to come to Turkey?'
"I have to think about it because Galatasaray is a great club with a positive approach."