The Red Devils suffered their third successive Premier League defeat at the Britannia Stadium after a 2-0 loss to Stoke City on Boxing Day.
The 20-time English champions have failed to win their last five Premier League games, prompting talk of Van Gaal being sacked.
Axed Chelsea manager Jose Mourinho is the favourite to replace the 64-year-old if United lose patience with the Dutch coach.
However, former Wales international Savage believes Giggs is the best candidate to take charge of the 20-time English champions until the end of the season.
If van gaal goes , giggs until end of season ?
— Robbie Savage (@RobbieSavage8) December 26, 2015
United are in fifth place in the Premier League table after their fifth defeat of the season.
Van Gaal has failed to win any silverware over the past 18 months although he did lead United back into the Champions League last season.
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