Van Aanholt played under Palace manager Sam Allardyce last season, as he kept Sunderland up.
The 26-year-old left-back, capped six times by his country, is the Premier League's top goal-scoring defender over the past two seasons, with seven goals.
"I want to keep all my good players and build," said Moyes.
Van Aanholt joined Sunderland from Chelsea, where he developed through the youth system, in July 2014.
The Black Cats are 18th in the table on 15 points, one behind Palace.
Former England manager Allardyce said on Thursday that his "interest is in strengthening the defence" at his new club.
"We have got bids in for three or four players and will find out if they are successful later on Thursday," he said.
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Sunderland have rejected a bid from Crystal Palace for their Netherlands defender Patrick van Aanholt, Black Cats manager David Moyes has confirmed.