The former Oranje striker, who has been Van Gaal's assistant since August 2012, had reportedly been set to link up with the 62-year-old at United.
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The Dutchman is expected to be named the next United manager this week after agreeing a deal to replace David Moyes 10 days ago during a meeting with a United delegation at his villa in the Algarve.
Giggs, who will take charge of United against Hull City on Tuesday night in the third of his four matches in interim charge, has spoken of how his own future remains unclear.
However, the obstacles to him being given a senior role on the coaching staff now appear to be less prominent than thought.
Danny Blind, Van Gaal's No.2 with the Netherlands, will also not be joining his fellow Dutchman at Old Trafford.
The delay in confirming Van Gaal's appointment is believed to have been partly due to extended negotiations over the make-up of the backroom team who will work under the former Barcelona, Bayern Munich and Ajax manager.
United senior officials are keen to keep Giggs and the 'Class of 92' on board, well aware of not only the need to plan for the future but also of their political importance to the club and its supporters.
Van Gaal, who turns 63 in August, represents a relatively short-term fix and a considerable departure from the long-term planning trumpeted by United in the wake of David Moyes's appointment 12 months ago.
Moyes's disastrous tenure came to an abrupt end when he was sacked a fortnight ago, and United chiefs swiftly put their faith in an experienced manager with a proven track record of success.
Van Gaal had three meetings with Tottenham but, as interest from Old Trafford grew, told the Spurs hierarchy that he was not interested in taking over at White Hart Lane.
He was also approached by Arsenal due to uncertainty over the future of Arsene Wenger, but has decided to take on the challenge of restoring United to pre-eminence.