The 19-year-old made his first-team debut for the Red Devils against Crystal Palace in the EFL Cup at the end of September, before featuring against Aston Villa in the Premier League on December 26.
Gore has been a regular in the first-team squad this term, but it has been decided that his development would be best-served by spending the second half of the 2023-24 campaign on loan.
Port Vale have won the race for his signature, and the teenager could be in line to make his debut for the club in Saturday's home fixture with Portsmouth in League One.
"Dan Gore has completed a loan move to Port Vale, which will run until the end of the season. The young midfielder leaves the Reds for a first spell elsewhere, and will gain experience of League One football with Andy Crosby's men," read a statement on Man United's official website.
"I am absolutely delighted to welcome Dan to Vale Park. Dan is a player that owns fantastic technical ability and the required competitiveness to his game to come in and immediately bolster the competition and options within our midfield," Port Vale head coach Crosby told the club's official website.
"As well as his obvious qualities on the pitch, Dan possesses all of the non-negotiable personal qualities that are required to be a member of the squad here at Port Vale.
"He has a real desire to make the most of his opportunity here and to learn from the experienced players within our squad to push himself beyond his limits to become the very best version of himself.
"I would also like to thank Manchester United for entrusting us with one of their elite young talents. With the opportunities that were afforded to all of our loan players, as well as our style of play and player care, both Dan and Manchester United see Port Vale as the perfect home to continue his already impressive development.
"We all look forward to working closely with Dan both on and off the pitch between now and the end of the season."
Meanwhile, Port Vale's director of football David Flitcroft added: "Firstly, I would like to thank Manchester United, Darren Fletcher, Les Parry and Dan's family for working with us to get this loan move agreed and secured.
"Dan has been on our radar for a while now; and when speaking to Dan and his family you can quickly understand what his cause is. His family are everything to him and he plays for them to make them proud. That is his driver.
"Dan possesses the quality on the ball to drive the team forward, but also has a real combative side to his game. He has a great maturity on and off the pitch that is ahead of his years, as well as a real desire and work ethic to push himself to become the very best player and person that he can be.
"Dan's signature was highly sought after by teams at this level and above. For Dan and Manchester United to choose us as a home for him speaks volumes of the work that we are doing at the club with our loan strategy.
"We have worked tirelessly to create trust and bonds with the country's best clubs and we can now present evidence of the strategy through our successful loans in the first half of the season, highlighting game time opportunities, a platform to learn and develop and support systems on and off the pitch.
"We as a club have been scouted by Manchester United over a period of time, they have interviewed us and selected us based on how we play and how we support young players. We all look forward to watching Dan's game evolve, and being part of his and our football journey."
Gore made the move to Man United from Burnley in 2018, and he helped the club's Under-18s win the FA Youth Cup during the 2021-22 campaign.