The striker is Van Gaal's captain for the Netherlands and reports have suggested that he will be given the armband at Old Trafford, if his fellow Dutchman becomes the club's new manager.
Wayne Rooney has been tipped for the captaincy following the departure of Nemanja Vidic to Inter Milan, but Van Persie is also a candidate for the role.
"He is my captain, he's my top scorer in the one-and-a-half years that he has played under me," Van Gaal told Sky Sports News.
"He became the all-time top scorer of the Dutch team, he plays superb football and even coming back from a bad injury he scores a fantastic, unbelievable goal. I was happy with his performance, but he is also a fantastic captain.
"I think you always make a player captain when you have the same morals and philosophy. Not only about football and tactics and what is happening on the pitch but about life."
Van Persie scored as the Netherlands drew 1-1 with Ecuador in a friendly on Saturday.