The 20-year-old, who joined Rotherham on loan last winter and recorded an assist on his debut, had a promising start.
However, he picked up the injury during training and has returned to Carrington for treatment.
United stated: “The academy will work closely with Rotherham United to support Dan to return as soon as possible."
Gore has also issued a statement, adding. “Having loved the start to my time here and been made to feel so welcome by everyone at the club, I am absolutely gutted to have picked up an injury.
“Everyone who knows me knows that I would play through any pain if I physically could, but unfortunately, I need to take a few weeks out. I’ll be giving everything to get back as soon as possible to contribute to the team and be able to play in front of the fans again.”