The 20-year-old has emerged as one of the most in-demand players in Scotland, having only signed for United from amateur side Queen's Park last summer.
Robertson quickly established himself at Tannadice and even become a full Scotland international under Gordon Strachan.
West Ham have tried to broker a deal, with their second offer understood to have been £1.5million, but United, who have just sold Ryan Gauld to Sporting Lisbon, rejected their bids.
Speaking to Sky Sports News, United chairman Stephen Thompson refused to be drawn on the subject.
"I'm not going to comment on that," he said. "I prefer to comment when business is done. If they prefer to comment, that is fine, but I am saying no more at the moment."
Everton, West Brom, Sunderland and Newcastle have all been credited with an interest in Robertson in recent months.