The 18-year-old made his debut in the mini Manchester derby at the Etihad in March, where he made a series of spectacular saves to help the Red Devils Under-18 side secure a 2-2 draw.
Richardson's profile on United's website reads, 'signed on a permanent contract after arriving on a short-term basis towards the end of the 2016-17 season'.
'One of four keepers with the reserves squad at the training camp in Austria — with Kieran O'Hara, Max Johnstone and Ilias Moutha-Sebtaoui — and likely to push for a place in the Under 23s and UEFA Youth League squad.'
'Formerly with Leeds United, he produced some outstanding performances when thrust into the Reds' U18s team to earn a deal over the course of the summer.'
Richardson will now represent United in the prestigious Super Cup NI tournament, in Northern Ireland this week.