The 39-year-old, who retired from playing last year, returned to Old Trafford to assist interim manager Ryan Giggs in the final four games of the Premier League season following David Moyes's sacking in April.
While Giggs has been installed as Van Gaal's right-hand man for the 2014-15 campaign, Scholes is not expecting to stay on with the 'Class of '92'.
In his blog for Paddy Power, Scholes wrote: "I've not spoken to [executive vice chairman] Ed Woodward. I came back for Ryan Giggs for the last few games to try to help out but I'm not waiting for a phone call and don't expect to be at Old Trafford next season."
Van Gaal agreed a three-year deal to become Moyes's successor on Monday.