While the 19-year-old is highly regarded at Old Trafford, the academy graduate has made just three starts and four substitute outings in all competitions.
With the last of those appearances coming in December, United are keen to find regular first-team football for Garner ahead of the next campaign.
Despite the departure of long-serving boss Eddie Howe, the managerless Cherries are said to be in the driving seat for Garner.
However, Swans boss Steve Cooper is familiar with Garner after his spell as coach of England Under-17s. body check tags ::