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.
The ex-Manchester United striker had been linked to a return to England and Stoke City have already made an official approach for him.
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.”