The 21-year-old - who has played once so far for the Magpies' first team, setting up a goal for Joelinton on his bow in the FA Cup against Rochdale in 2020 - follows Oisín McEntee in making a temporary move to the Scottish Championship side.
He heads to Cappielow until the end of the 2021/22 campaign.
Allan, who was involved with Steve Bruce's senior squad in pre-season, played for 80 minutes of United's Papa John's Trophy tie against Sheffield Wednesday on Tuesday before completing his move to the Ton.
It is a second loan spell for the youngster after he spent the first part of last season with Accrington Stanley.