Goodwillie, who made his Scotland debut against the Faroe Islands in November, has scored 11 goals this season, attracting attention from clubs in England and Scotland.
Dundee United chairman Stephen Thompson told STV: "There have been two formal approaches to me from a chairman and a chief executive of two other clubs, but there's been no offers for David.
"There's been a lot of enquiries, a lot of interest shown but no offers to date so he remains a Dundee United player as far as I'm concerned. Obviously he's in great form and has a great future ahead of him. We'll see what happens in the next few weeks."
Manager Peter Houston last week stated he hoped to keep his squad together, but Thompson said United may sell during the January transfer window.
The chairman added: "Let's be fair, every club is a selling club. If the biggest clubs in the world like Manchester United sell Ronaldo then anybody can sell anybody. Every player has a price and that's just the way football is. We'll see in the weeks to come if we get any firm offers and that's it."