Norkett, 17, is currently in his second season with The Red Devil's Academy side.
The striker becomes the first player since 1907 to tread the path from Old Trafford to the Northolme, back then Tommy Arkesden made the switch.
"We are very thankful and grateful to Manchester United and Travis Binnion (Lead Professional Development Phase Coach) for giving us the opportunity to have Manni with us for the coming weeks,” said Shaw.
“Their Academy Manager, Nick Cox, is one of the best around and he is always keen to give his young players experience in adult football when appropriate and we are fortunately the chosen club for Manni.
“It is testament to Gainsborough Trinity and the club's reputation that we are being trusted for a period of Manni's development and to have a highly rated young player from a huge football club joining us, it shows what people are thinking about our work this year and the environment we have created."
"Manni plays with plenty of confidence and authority for such a young man and shows a real humility with how hard he worked. It will be an eye-opener for him when he takes the pitch at this level, but I feel we have the pleasure of a real prospect in our squad for a handful of games."
On the pitch a depleted Trinity side picked up a 1-1 draw at Grantham on Monday after Jovon Makama’s 83rd minute goal cancelled out a first half opener by Jonathan Wafula.
Trinity were missing a couple of players due to COVID, along with management duo Tom Shaw and Damon Parkinson.
They are next due to be in action at home to league leaders Matlock Town on New Years Day.
That is then followed by a trip to Witton Albion the following Saturday.