The 16-year-old midfielder burst onto the scene during the first weekend of the current Premier League season when Hammers chief Slaven Bilic opted to start him against Arsenal.
The rookie shone at the Emirates Stadium as the Irons sealed a memorable win, with his shackling of World Cup-winning playmaker Mesut Ozil particularly notable.
He has since dipped away from the West Ham starting XI as he continues his development, however, reports claim that United are very much interested in the youngster.
The Manchester heavyweights are expected to make an £8m offer next month, although their London counterparts are likely to resist.
As well as United, Liverpool, Manchester City and Chelsea are all interested.
The same report suggests that United also tried to sign Oxford during the summer just gone, prior to his breakthrough.