The 18-year-old embarks on his first temporary move away from the club having made his first-team debut as a last-minute substitute, in the 2-1 Premier League win at Wolves in May 2021.
Fish will remain with United for the beginning of pre-season before joining up with the Hatters at an agreed time ahead of their National League campaign.
The England Under-18 international has spent the last week preparing for the new season with the first-team at the Reds' training camp in Surrey.
This is the second Academy loan of the summer transfer window with individual player development again being prioritised across the Professional Development Phase.