The Red Devils had been hoping to claim an unprecedented fourth successive title but have stumbled of late.
Their bid took a major dent over the weekend as they drew with Blackburn, meaning leaders Chelsea could go four points clear on Tuesday as they play Bolton in their game in hand.
Whilst Giggs admits that United are no longer favourites, he insists the title race is not yet over.
"It is certainly in Chelsea and Arsenal's hands now," he told Sky Sports News.
"We dropped points at Blackburn which was disappointing but we will keep fighting to the end and hopefully Arsenal and Chelsea will drop points and we will capitalise."
Giggs is now looking ahead to Saturday's Manchester derby against rivals City, adding: "It was always going to be huge at this time of the year, they are going for fourth and we are going for the championship and derby days anyway are massive.
"There is a bit of extra spice to this game but we have faced them three times this year and they have all been great games so hopefully that will be the case again with us coming out on top."
Giggs was forced off at Blackburn with injury but he is hopeful of facing City.
"I strained my hamstring against Blackburn. I felt it in the first 20 minutes but the scan showed nothing so I should be okay," he said.