The Dutchman has come under significant pressure in recent weeks, with reports suggesting that he had offered to resign following Saturday’s 1-0 defeat to Southampton.
Giggs has been mentioned as a potential successor to Van Gaal, with Jose Mourinho also thought to be in the frame.
According to the newspaper, the United board are worried that Giggs could end his 28-year association with the club and walk away if the job is given to former Chelsea boss Jose Mourinho as Giggs would be reluctant to continue as a No 2.
Giggs is also reportedly reluctant to be caretaker boss again, unless it leads to a permanent job, something that Ed Woodward is afraid to do.
Woodward and United are in a tough position here. The best manager on the planet, Pep Guardiola is taking over at cross-town rivals Manchester City next summer, where based on his track record, he will win trophies.
Jose Mourinho is arguably one of the top three managers in the world and he is available. If United hire the Special One, you would expect United to battle City for trophies at home and in Europe next season.
But Mourinho, as is Guardiola, is a short-term fix. He will last three seasons then leave. Then what?
There is a sizable contingent of the Manchester United fans who would like to see Giggs, who understands the United way to be given the chance. They point to the success that Guardiola, former Barca star, had when he took over as their manager. And look at Real Madrid, where another footballing legend Zinedine Zidane has taken over.
United fans sing about playing football the Matt Busby Way. Giggs, who won 34 trophies as a player, will get United playing that way again. Mourinho will not. But Mourinho will probably win more trophies in the short-run.
What should the United board do? Giggs or Mourinho?