The 20-year-old left-back had also been linked with West Ham and West Brom, but Hull City beat off competition from their Premier League rivals, securing the youngster with an improved offer thought to be in the region of £3million.
United boss Jackie McNamara had been keen to retain Robertson, who had only been at Tannadice for a year, joining from amateur League Two side Queen’s Park last July.
But he could not stand in the way of a switch to Hull, who are expected to finalise the deal next week.
Robertson has made 44 appearances for United in all competitions, scoring five goals, and has been capped twice for Scotland with his most recent appearance coming in the 2-2 draw against Nigeria in May.
He is to become Hull's latest summer signing and will join fellow recruits Jake Livermore, Thomas Ince and Robert Snodgrass.